{"cells":[{"cell_type":"markdown","metadata":{"id":"C2B9935EF857448FAA738AACD538D101","jupyter":{},"notebookId":"68f9c684c011c75553f024e8","runtime":{"status":"default","execution_status":null,"is_visible":false},"scrolled":false,"slideshow":{"slide_type":"slide"},"tags":[]},"source":" # <center> Lecture7 : Posterior Inference &  Estimation  </center>  \n \n ## <center> Instructor: Dr. Hu Chuan-Peng  </center>"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"16336C353BD4413B9BFED47E2D0B902E","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"前面的课程中，我们介绍了如下几个知识点：  \n- 贝叶斯推断的“精神”: 根据数据来更新对某个未知参数的信念(posterior distribution);  \n- 贝叶斯公式的四部分：后验、先验、似然、分母(归一化因子)  \n\n$$  \nP(A|B) = \\frac {P(A)P(B|A)}{P(B)}  \n$$  \n\n- 贝叶斯推断的难点：获得后验  \n\t- 解析解(套公式): conjugate  \n\t- 近似解: 网格法、**MCMC**  \n- MCMC的原理与实现  \n\t- Stan  \n"},{"cell_type":"markdown","metadata":{"id":"C5B6C1BC7B66412C94C5D273137A97D5","notebookId":"68f9c684c011c75553f024e8","runtime":{"status":"default","execution_status":null,"is_visible":false},"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"}},"source":"- Stan是概率编程语言（PPL）中的一种，可帮助我们进行贝叶斯模型的构建和后验推断。  \n- 在Stan中，定义先验分布、似然函数（模型）后，通过其采样器(sampler)来获得MCMC的采样。  \n- Stan 支持多种采样方法，如 MCMC（马尔可夫链蒙特卡洛），这使得 Stan 在保持高效性的同时，能够处理更复杂的模型，满足多种统计建模需求。  \n- Stan 是 PPL 框架的一个代表，不同的 PPL 框架（如 PyMC）在设计和功能上各有特色。"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"B98AFEA757204A1FB5A781D3A1255132","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"## 回顾：概率编程语言（PPL，probabilistic programming language）  \n\n概率编程语言的核心作用在于自动帮助我们进行后验采样，获得贝叶斯后验分布。  \n\n传统贝叶斯后验分布的推理中，我们不仅需要自己定义模型（似然函数、先验），还需要想办法获得后验如（MCMC）。  \n\n```{R}  \nN <- 10; Y <- 9  \nx <- seq(0, 1, length.out = 10000)  \n\npi <- dbeta(x, shape1 = 2, shape2 = 2)  \n\nlikelihood <- dbinom(Y, size = N, prob = x)  \n\nposterior_unstandardized <- likelihood * pi  \n\n# then we need MCMC to get the posterior distribution  \n# each iteration we will calculate posterior_unstandardized = likelihood * pi  \n\nmh_simulation <- mh_tour(N=2000)  \n```\n\n<div style=\"padding-bottom: 30px;\"></div>"},{"cell_type":"markdown","metadata":{"id":"0278C38FE51E45CDA0DFB7579FF9C8E1","notebookId":"68f9c684c011c75553f024e8","runtime":{"status":"default","execution_status":null,"is_visible":false},"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"}},"source":"## 回顾：概率编程语言（PPL，probabilistic programming language）  \n\n在概率编程语言框架下，只需要定义模型（似然函数、先验），然后可以让这些工具帮助我们获得后验。  \n\n```{R}  \n# 定义模型  \nstan_model_code <- \"  \ndata {  \n  int<lower=0> N;       \n  int<lower=0, upper=n_trials> Y;  \n}  \nparameters {  \n  real<lower=0, upper=1> pi;  \n}  \nmodel {  \n  pi ~ beta(2, 2);  \n  Y ~ binomial(N, pi);  \n}  \n\"  \n\n# 拟合模型  \nfit <- stan(  \n  model_code = stan_model_code,      # 定义的模型或模型文件路径  \n  data = stan_data,                  # 输入数据  \n  chains = 1,                        # 马尔可夫链数量  \n  iter = 2000,                      # 总迭代次数（每个链）  \n  warmup = 1000,                     # 热身迭代次数（不保存）  \n  cores = 2                          # 使用的CPU核心数  \n)  \n```\n\n可见，我们只需要在 Stan 的 `model` 中定义似然函数模型之后使用stan的sampler就可以自动计算出后验分布, 而无需考虑如何使用 MCMC 等方法来计算后验分布。  \n- 其中  `Y ~ binomial(N, pi)` 相等于公式 $likelihood = \\text{Binomial}(\\text{N}, \\text{Y} | \\pi)$。  \n\n<div style=\"padding-bottom: 30px;\"></div>"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"9CDD8B8554214065B0C7AD43690FCF01","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"\nPPL 框架以及当前的贝叶斯推断的生态提供了其他更多的功能和特点，后面的课程中将接触到：  \n\n\n| 功能           | 描述                                                          | 主要用途                      |  \n|-------------------------|---------------------------------------------------------------------------------------|--------------------------------------------------------|  \n| **模型定义**            | 用简洁语法定义先验分布和似然函数，支持分层模型   | 简化模型构建流程，灵活定义复杂结构        |  \n| **自动采样**            | 提供 MCMC (HMC、NUTS)、变分推断等多种近似算法     | 高效获取后验样本，适应高维复杂模型           |  \n| **自动微分**            | 自动计算梯度，支持高维复杂模型                | 加速模型优化，适应复杂计算需求                    |  \n| **诊断与调试**          | 提供 Trace Plot、R-hat 等诊断工具，检测采样质量与收敛性       | 确保采样结果可靠，识别采样问题                |  \n| **后验分析和可视化**    | 提供后验分布图、密度图、置信区间等可视化工具     | 直观理解参数分布和不确定性，辅助决策             |  \n| **后验预测**            | 从后验分布生成新观测数据，用于模型验证       | 检查模型合理性，评估预测效果               |  \n| **灵活建模扩展性**      | 支持模块化建模、集成外部工具，构建自定义模型        | 适应不同应用场景，增强模型的扩展性         |  \n"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"C218E1C1D5C648FD96495E319EF63D50","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"### 贝叶斯推断（概率编程语言）的核心术语  \n\n这些术语在贝叶斯统计和概率编程中具有重要应用。除了已经讲过的部分概念外，本节课将涉及一些新概念：  \n\n\n| 术语（中英文全称）                                     | 定义                                                                                  |  \n| --------------------------------------------- | ----------------------------------------------------------------------------------- |  \n| 先验分布（Prior Distribution）                      | 先验分布是指在观察数据之前，研究人员对模型中未知参数的初始信念。这种信念可以基于现有研究或试点数据形成。                                |  \n| 似然函数（Likelihood Function）                     | 似然函数是指特定参数下观测数据的概率，它是统计模型中选择参数的概率函数。例如，伯努利函数是描述硬币投掷统计的似然函数。                         |  \n| 后验分布（Posterior Distribution）                  | 后验分布是指在观察到数据后，根据贝叶斯规则平衡先验知识与观测数据，对参数的更新信念。                                          |  \n| 马尔可夫链蒙特卡洛方法（Markov Chain Monte Carlo, MCMC）   | MCMC是一种通过模拟来推断后验分布的采样方法。通过算法构建多个马尔可夫链，使得它们的平稳分布与感兴趣的后验分布相匹配，这个过程称为MCMC收敛。           |  \n| 采样样本（Samples/Draws）                           | 采样样本值是指从后验分布中抽取的参数值，用作后验分布的近似，并通过蒙特卡洛积分获得后验分布的经验估计和感兴趣的汇总统计量。其采样结果在计算机术语中也称为 trace。 |  \n| 调试样本（Tune）                                    | 调试样本是指在MCMC抽样过程中用于达到平稳分布的初始阶段，此阶段的样本通常被丢弃，不在最终分析中使用。也叫做 Warmup 活 burn-in，燃烧样本。      |  \n| 链（Chains）                                     | 链是指来自单个MCMC链的一系列样本（或抽样值）。链用于诊断收敛性以及程序在特定应用中的其他潜在问题。                                 |  \n| 有效样本大小（Effective Sample Size, ESS）            | 有效样本大小是指与N个自相关样本具有相同估计力的独立样本数量。ESS常用于判断MCMC链中的抽样数量是否足以保证可靠的估计不确定性。                  |  \n| Gelman-Rubin统计量（Gelman-Rubin Statistics, R̂）  | Gelman-Rubin统计量是链内变异性与链间变异性的比率。所有参数和感兴趣量接近1.0的值表明马尔可夫链蒙特卡洛算法已足够收敛到平稳分布。             |  \n| 最高密度区间（Highest Density Interval, HDI）         | 最高密度区间是贝叶斯统计中参数可信范围的估计。它包括后验分布中的一个区间，该区间内的每个点的密度都高于区间外的点。                           |  \n| 实践等效区间（Region of Practical Equivalence, ROPE） | 实践等效区间是预先定义的参数值范围，这些值被认为在实践上等同于零。用于判断参数估计是否显著不同于零。                                  |  \n| 贝叶斯因子（Bayes Factor, BF）                       | 贝叶斯因子量化了一个统计模型相对于另一个模型的证据强度。大于1的值表明相对于原始模型，替代模型得到了更多的支持。                            |  \n\n"},{"cell_type":"markdown","metadata":{"id":"24E563A9EE1E42A4BF5465DBE970B476","notebookId":"68f9c684c011c75553f024e8","runtime":{"status":"default","execution_status":null,"is_visible":false},"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"}},"source":"## 回顾推断统计  \n- 学习统计的目的是什么？  \n\t- 参数估计；  \n\t- 假设检验；  \n\n- 获得后验之后，接下来要考虑与推断统计的目标进行整合。  \n"},{"cell_type":"markdown","metadata":{"id":"6BBF17A1FA0941218598833E45521119","jupyter":{},"notebookId":"68f9c684c011c75553f024e8","runtime":{"status":"default","execution_status":null,"is_visible":false},"scrolled":false,"slideshow":{"slide_type":"slide"},"tags":[]},"source":"## 利用后验分布进行统计推断  \n\n⭐ 通常，通过 Stan 等概率编程语言(PPL) 得到后验之后，需要首先确保MCMC的后验是有效的，然后进行推断。  \n\n本节课将主要包括：  \n* MCMC诊断  \n* 后验估计  \n* 假设检验  \n\n\n\n<p align=\"center\">  \n  <img src=\"https://cdn.kesci.com/upload/sm7veon5ei.png?imageView2/0/w/720/h/960\" alt=\"Image Name\">  \n</p>  \n"},{"cell_type":"markdown","metadata":{"jp-MarkdownHeadingCollapsed":true,"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"2096E9990B4D4AAF8ED041BF28AB640F","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"\n### 以Beta-Binomial模型在Stan中的实现为例  \n\n仍然以随机点运动任务为例，每个试验中参与者判断正确的概率用 $\\pi$ 表示。  \n\n**模型假设**  \n\n- 判断正确概率 $\\pi$ 服从 **Beta 分布**分布：  \n\n$$  \n\\begin{equation}  \n\\pi \\sim \\text{Beta}(\\alpha, \\beta)  \n\\end{equation}  \n$$  \n\n- 在每次试验中，参与者的成功次数 $Y$ 服从 **Binomial 分布**：  \n\n$$  \n\\begin{equation}  \nY \\sim \\text{Binomial}(n, \\pi)  \n\\end{equation}  \n$$  \n\n其中，$n$是试验的总次数，$Y$是成功次数。  \n\n**模型设定**  \n\n在这个例子中，我们可以使用Beta-Binomial 模型来表示：  \n\n> 总试验数为n=10，成功次数为9 $(Y = 9)$  \n> 先验分布为：$\\pi    \\sim \\text{Beta}(2, 2)$  \n> 似然函数为：$Y|\\pi  \\sim \\text{Bin}(n, \\pi)$  \n\n对应的 Rstan 代码为：  \n```R  \n# 准备数据  \nstan_data <- list(N = 10,Y = 9)  \n\n# 定义模型  \nstan_model_code <- \"  \ndata {  \n  int<lower=0> N;       \n  int<lower=0, upper=n_trials> Y;  \n}  \nparameters {  \n  real<lower=0, upper=1> pi;  \n}  \nmodel {  \n  pi ~ beta(2, 2);  \n  Y ~ binomial(N, pi);  \n}  \n\"  \n```\n\n<div style=\"padding-bottom: 30px;\"></div>"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"1A4058474F79499A8402CC8F76C3AC43","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"## 对马尔科夫链的诊断 (Markov chain diagnostics) "},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"00B62FFC98F64BF4BC16772CFF924FB0","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"当定义好 Stan 模型后，就可以使用rstan包中的 `stan()` 函数进行采样  \n在下面的例子里，采样的结果都保存在 `fit` 中，  \n可以使用rstan包中的函数直接对结果进行可视化，  \n也可以使用 `extract() ` 函数来获取参数 pi 的后验分布的采样并单独保存，以便进一步的分析。  \n\n```R  \nfit_bb <- stan(  \n  model_code = stan_model_code,      # 定义的模型或模型文件路径  \n  data = stan_data,                  # 输入数据  \n  chains = 4,                        # 马尔可夫链数量  \n  iter = 2000                        # 总迭代次数（每个链）  \n)  \n\n# m_bb <- rstan::stan_model(file = stan_model_code)  \n\n# fit_bb <- rstan::sampling(  \n#  object = m_bb,  \n#  data = data,  \n#  iter = 2000,  \n#  warmup = 1000,  \n#  chains = 4,  \n#  seed = 123,  \n#)  \n\nidata <- rstan::extract(fit_bb)  \nsample_pi <- as.data.frame(idata$pi)  \n```\n\n但使用这些采样结果前，我们需要回答一个问题：这些采样结果是否可信？🤔"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"2CAC73A21B0B4B4CAAF4806DAC2DFD7E","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"由于这些采样来自于 MCMC 算法，即马尔科夫链蒙特卡洛算法，而模拟采样方法的关键是 \"近似 \"和 \"收敛\"  。  \n\n这就引出了以下问题：  \n\n- 好的马尔可夫链是什么样的？  \n- 如何判断马尔可夫链样本是否产生了合理的后验近似值？  \n- 马尔可夫链样本量应该多大？  \n\n诊断的意义：确定好的trace的标准，为我们不知道真实的后验时作为标准进行推断。"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"AA0C53DFD9134646894A40367F6DF68E","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"在本节中，将重点介绍几种诊断方法：  \n\n- 可视化诊断：轨迹图 (trace plot)和平行链 (parallel chains)。  \n- 量化诊断指标：有效样本大小 ESS(effective sample size)、自相关性 (autocorrelation)和 $\\hat{R}$ (R-hat)。  \n    \n> 这些诊断方法的使用应综合考虑。没有一种视觉或数值诊断方法是万能的，通常要一起考虑。"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"1134E6FEB0FE431F9302CB1CB948BA96","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"### 轨迹图 (trace plot)  \n\n轨迹图 (trace plot) 是一种可视化方法，用于显示马尔可夫链的轨迹，即马尔可夫链在参数空间中的位置。  \n- 图中的横坐标是时间步长，纵坐标是参数的值。  \n- 轨迹图往往结合参数的后验分布一起展示，以便于观察采样样本对于参数分布的代表性。"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"9842C2D257B84CA19CEE6F3F3956E50E","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"我们先来看一下正常的轨迹图是什么样：  \n\n\n![Image Name](https://cdn.kesci.com/upload/s2le92xkaq.png?imageView2/0/w/500/h/500)  \n\n- 这是之前 Beta-Binomial model 的示例。  \n- 上图：轨迹图看起来像一堆白噪声，没有明显的趋势或现象，这意味着**链是稳定的**。  \n- 下图：展示了正常的参数后验分布的情况。  \n\n<div style=\"padding-bottom: 30px;\"></div>"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"3BCA5BD46B5140BB94D6638286BF1065","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"\n![Image Name](https://cdn.kesci.com/upload/s2le9nbz6q.png?imageView2/0/w/450/h/450)  \n\n\n显然这种条件下的后验参数分布在用于后续推断时是必须非常谨慎。  \n\n图中上部分链 A：  \n\n- 链 A 中的轨迹图显示，它在 5000 次迭代后还没有稳定下来，并且它只探索了 0.6 到 0.9 之间的参数值。  \n  - 下降趋势也暗示着链值之间存在着很强的相关性--它们看起来并不像独立的噪音。  \n  - 虽然马尔可夫链本身具有依赖性，但它们越像噪声(独立样本的表现)，得出的后验近似值误差就越小（粗略地说）。  \n- 结合后验分布图：**它的后验近似值高估了真实分布中央部分，而完全低估了这个范围之外值的可信度**。  \n\n图中下部分链 B：   \n- 链 B 表现出不同的问题，迹线图中的部分区域存在**两条完全·平直的线**所显示的那样，  \n  - 这意味着，**当它采样到较小的参数值时，往往会陷入这个值的附近，** 这也表明了一种局部的高相关性。  \n  - 链条 B 在陷入困境时，会对后验参数左侧尾部的值进行**过度采样**。         \n- 结合密度图  \n  - 虽然链 B 后验分布和真实分布的重合性更好，但它存在多峰分布，原因在于过度采样。  \n\n<div style=\"padding-bottom: 30px;\"></div>"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"F84C553018AB40999053D438680B1C7D","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"如果我们得到一个糟糕的轨迹图，如何进行补救📍：  \n\n- 1. 检查模型。确定假定的**先验模型**和**数据模型**是否合适？    \n- 2. 对数据链进行**更多迭代**。一些不理想的短期连锁趋势可能会在长期内得到改善。"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"86FA246E72C34718935BF97DFD3C9AB8","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"### 有效样本大小 ESS(effective sample size)和自相关性 (autocorrelation)  \n\nMCMC 链的样本存在相关性，后一个参数的采样过程与前一个参数有关，这带来的问题是仅间隔一个位置的参数样本间的相关性很高。  \n\n然而，MCMC 链又要求“无记忆性”，即样本之间最好是独立的。  \n\n🤔那么如何理解 MCMC链中独立与相关的矛盾？  \n- 需要清楚的是，相关性仅限于间隔一个位置的参数样本。  \n- 而独立性的要求是针对于参数样本间大于一个间隔的情况的。  \n\n自相关就可以很好的描述样本之间的相关性；而有效样本量针对于描述样本的独立性。"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"51011C78E3594D60A67AD5E85F664E0D","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"**自相关性 (autocorrelation)**  \n\n- 可用于评估马尔科夫链的样本之间的相关性。  \n- 强烈的自相关性或依赖性是一件坏事--它**与较小的有效样本比相伴而生**，说明我们得出的后验近似值可能不可靠。  \n    \n根据马尔可夫链的简单构造，链值之间必然存在一些自相关性--一个链值($\\pi_{i}$)取决于前一个链值（$\\pi_{i-1}$），而前一个链值（$\\pi_{i-1}$）又取决于前一个链值（$\\pi_{i-2}$），依此类推。**这种依赖链还意味着，每个链值都在一定程度上依赖于之前的所有链值**。  \n\n- 例如，$\\pi_{i}$ 取决于$\\pi_{i-1}$，而$\\pi_{i-1}$ 取决于$\\pi_{i-2}$，因此$\\pi_{i}$ 也取决于$\\pi_{i-2}$。  \n- 然而，**这种依赖性或自相关性会逐渐消失**。  \n    \n> 这就像托布勒的地理学第一定律：万事万物都与其他事物相关，**但近处的事物比远处的事物更相关**。"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"C29F57B22C2A426EA858C4A276612B81","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"一个自相关良好的例子：  \n\n\n![Image Name](https://cdn.kesci.com/upload/s2lefvdivs.png?imageView2/0/w/640/h/640)  \n\n- 图左是之前提到的轨迹图(trace plot)。  \n- 右图是自相关图，x轴是样本之间的间隔距离，y 轴为自相关性。  \n- 右图x轴0处：所有参数采样和自己的相关性 (这显然为1)；  \n- 右图x轴1处：所有参数和间隔一个距离参数的样本的相关性，该相关性依然很高，大约为 0.5，表明相差仅 1 步的链值之间存在中等程度的相关性。  \n- 随后自相关性迅速下降，到间隔 5 个样本时相关性下降为 0。  \n- 可以推断：间隔越大，马尔科夫链值之间的相关性越小。说明马尔科夫链正在**快速混合**，即在后验可信 π 值范围内快速移动，从而至少模拟了一个独立样本。  \n- 对于间隔解释的例子：如果有100个参数形成的样本，编号为 1-100。 那么间隔10，就代表，10-100组成的样本。为了让 10-100 和原始的样本长度相同，我们可以使用 1-90和 10-100 的样本进行相关分析。需要注意的是，存在更好的数学方法可以避免间隔越大，样本量越小的问题。扩展阅读: https://zhuanlan.zhihu.com/p/77072803  \n  \n<div style=\"padding-bottom: 30px;\"></div>"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"FA8C53C1F320427395D8E01F1D570819","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"一个自相关糟糕的例子：  \n\n\n![Image Name](https://cdn.kesci.com/upload/s2lew3t7qx.png?imageView2/0/w/640/h/640)  \n- 自相关曲线的缓慢下降表明，链值之间的依赖关系不会很快消失。  \n- 相差整整 20 步的马尔可夫链值之间存在大约 0.9 的相关性！  \n- 由于其链值与前值的联系非常紧密，因此该链的**混合速度很慢**，**需要很长时间才能充分探索后验的全部范围**。  \n- 因此，应该谨慎使用该链来近似后验。  \n\n<div style=\"padding-bottom: 30px;\"></div>"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"D34FDD4D5C58419F90C6128E7575DECF","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"**重要概念：Fast vs slow mixing Markov chains**  \n\n- 快速混合：指链表现出与独立样本的行为，即混合链值之间的自相关性迅速下降，有效样本大小比相当大。  \n    \n- 慢速混合：指链不具有独立样本的特征，即混合链值之间的自相关性下降非常缓慢，有效样本大小比很小。换句话说，链在后验可信值范围内 \"缓慢 \"移动。"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"BD441A322202465F9E0768CECCF43F21","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"如果MCMC的混合速度太慢该怎么办呢？一些推荐的策略📍：  \n\n- 第一， \"运行更长的链\"。如果迭代次数足够多，即使是缓慢的混合链最终也能产生良好的后验近似值。  \n    \n- 第二，使得马尔科夫链更薄。  \n    \n    - 例如，在包含 5000 个样本的链中，我们每间隔一个参数来保留样本，并丢弃其余的样本，例如：{$\\pi_{2}$, $\\pi_{4}$, ..., $\\pi_{5000}$}.  \n        \n    - 甚至，我们可以每间隔十样本来保留样本：{$\\pi_{10}$, $\\pi_{20}$, ..., $\\pi_{5000}$}.  \n        \n- 通过舍弃中间的抽样，我们可以消除低间隔带来的强相关性。  \n    \n    - $\\pi_{20}$与稀疏链中的前一个值 ($\\pi_{10}$)的相关性小于与原始链中的前一个值 ($\\pi_{19}$)的相关性。"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"94BAD27725A34A2DB16C51D55945C089","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"**一个使 MCMC 链变得更薄(thin)来减少自相关的示例：**  \n\n我们使用 thin=10 来使得链变得比之前薄10倍，也就是从2000个样本中，每隔10个样本宝保留一个，总共200个样本将被保留。  \n\n\n![Image Name](https://cdn.kesci.com/upload/s2lew3t7qx.png?imageView2/0/w/640/h/640)  \n\n![Image Name](https://cdn.kesci.com/upload/s2lf7kf4e8.png?imageView2/0/w/640/h/640)  \n\n- 上图为之前没有变薄的MCMC链。下图为变薄了10倍的链。  \n- 可以看到，自相关性产生了快速的下降。  \n- 但是，这可能任然无法完全消除自相关性。  \n\n🤔一个值得思考的问题：  \n\n- 使用 thin=10 时，我们可用的样本变得更少，是否值得损失 90% 的原始样本值呢？  \n<div style=\"padding-bottom: 30px;\"></div>"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"5A7FDAF11DF14C6FBB75B67B157BC04F","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"\n**有效样本量比率(Effective sample size ratio)**  \n\n为了避免链中相关性的影响，研究者提出了有效样本量 (effective sample size)的指标，描述MCMC链中可视为独立样本的数量。  \n\n根据 MCMC 的特性，需要链存在无记忆性，也就是大于两个间隔的参数样本间不存在或者存在很小的相关性。有效样本量即是这些不存在或仅存在弱的相关性样本的数量。  \n\n$\\frac{N_{neff}}{N}$  \n\n- N 表示依赖马尔可夫链的实际样本量或长度。  \n- $N_{eff}$ 为马尔可夫链的有效样本大小，量化了产生等效精确后验近似所需的独立样本数量。  \n- $N_{eff}$ 越大越好，但马尔可夫链近似的准确性通常只能达到较小独立样本的准确性。也就是说，  \n- 通常情况下，$N_{eff}$ < N，因此有效样本大小比小于1。我们往往会对有效样本量比小于 0.1 的马尔可夫链产生怀疑，即 Neff/N < 10%。  \n\n假设马尔科夫链的长度为 20,000，并且包括 6800 个独立样本，即有效样本量 ESS = 6800。  \n- 因此，有效样本量比率 EFF=34%，代表相当于我们只使用了 34% 的独立样本。  \n- 由于这个比例高于 0.1，这是可以接受的。并且 ESS > 400。  \n  \n> 更详细的定义请参见 Vehtari 等人（2021）。如需了解有效样本量, R-hat等指标的的联系，请参阅 Vats 和 Knudson (2018)。"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"89836F04753F4132A04028FBE29F3419","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"**补充，如何计算ESS**  \n\n\nESS表示与独立和均匀采样相当的有效样本数量，可以使用以下公式计算：  \n\nESS = $N / \\tau$  \n\n- 其中，N是总的采样数量。  \n- $\\tau$为自相关时间。  \n\n估计自相关时间（Autocorrelation Time，$\\tau$）：自相关时间表示样本之间的相关性消失所需的步数。它可以通过截尾自相关函数进行估计。  \n自相关时间$\\tau$可以通过求和截尾自相关函数的面积来估计，可以使用以下公式计算：  \n\n$\\tau = 1 + 2 * \\sum{(TACF(lag))}$  \n\n- 其中，lag表示滞后的范围。  \n\n\n计算自相关函数（Autocorrelation Function，ACF）：  \n- 首先，计算采样序列中不同滞后（lag）的自相关系数。  \n- 滞后为0时，自相关系数等于1；滞后为1时，自相关系数表示第一个样本与第二个样本之间的相关性；以此类推。  \n- 自相关函数的计算可以使用统计软件或编程语言中的相关函数。  \n\n计算截尾自相关函数（Truncated Autocorrelation Function，TACF）：  \n- 为了避免计算无限滞后的自相关系数，通常会截取滞后一定范围内的自相关系数。  \n- 选择一个合适的截断范围，例如滞后至自相关系数小于0.1，可以避免过多计算。  \n"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"49E95A2BD04240EA8CD94FC1890F898A","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"### R-hat and comparing parallel chains  \n\n自相关和有效样本量仅针对于描述一条MCMC链的有效性。假如我们获得了四条并行的马尔可夫链。我们不仅希望看到每条链的稳定性，还希望看到**四条链的一致性**。  \n\n可以想象，如果四条链的结果并不一致，那么推断结果的可靠性就很难保证。  \n\n同样，我们首先展示一个良好的并行链的结果：  \n\n\n![Image Name](https://cdn.kesci.com/upload/s2lg1juesc.png?imageView2/0/w/640/h/640)  \n\n- 四条链产生的后验近似值几乎没有差别，这证明我们的模拟是稳定的。  \n\n<div style=\"padding-bottom: 30px;\"></div>"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"3CAED2318DDA4AE5B1E1B7A8970B8674","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"为了便于比较和展示一个糟糕的并行链的结果，我们用更短的马尔可夫链模拟来演示。  \n\n\n![Image Name](https://cdn.kesci.com/upload/s2lfz6esan.png?imageView2/0/w/640/h/640)  \n\n- 运行四条并行链仅进行 100 次迭代，其中 50 次为warmup (见之前的词典解释)，因此最终产生 50 + 100 个样本量。  \n\n- 虽然这些链的轨迹图表现出相似的随机行为，但它们对应的密度图却不尽相同，因此产生的后验近似值也不尽相同。  \n\n这表明，如果我们只进行了 100 次迭代就停止模拟，那结果是不稳的。  \n\n<div style=\"padding-bottom: 20px;\"></div>"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"90A928AF97F141F79D8C753A3C5689B7","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"此外，不同链样本之间的变异性的一致性非常重要。  \n\n之前的结果同样显示，其中四条链具有相似的变异：  \n\n![Image Name](https://cdn.kesci.com/upload/s2lg1juesc.png?imageView2/0/w/640/h/640)  \n\n\n- 上图中间显示，四条链的可变性几乎是相同的。  \n- 上图右显示，所有链的变异与综合为单个链的变异性相似。  \n\n<div style=\"padding-bottom: 10px;\"></div>"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"BC4E15CB40D8478EA0B13A16F307C0F9","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"一个糟糕的例子：  \n\n\n![Image Name](https://cdn.kesci.com/upload/s2lg5bgzbr.png?imageView2/0/w/640/h/640)  \n\n\n- 四条平行链产生了相互冲突的后验近似值（中图）  \n- 当我们将这些链组合在一起时，后验近似值不稳定且较差（右图）。  \n- 因此，不同MCMC链参数值的范围和变异性远大于任何单个链中参数值的范围和变异性。  \n\n<div style=\"padding-bottom: 10px;\"></div>"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"A4E6131ABB504645BD0722E549B2EA0C","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"可直观地看出，所有链条上的数值变化与单个并行链条内部的数值变化之间的关系非常重要。具体来说  \n\n- 在 \"好的 \"马尔可夫链模拟中，所有并行链的变异性总和将与任何单个链内的变异性**大致相当**；  \n- 在 \"坏的 \"马尔可夫链模拟中，所有并行链的变异性总和可能**超过**每个链内的典型变异性  \n    \n可使用 $\\hat{R}$ 量化综合链变异性和链内变异性之间的关系。"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"963938540CD745229126CBFB52199C16","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"$\\hat{R}$ 指标（称 \"R-hat\"）为并行链的可视化比较提供了具体的数学指标。  \n\n\n$$  \n\\hat{R} = \\frac{V_{combined}}{V_{within}}  \n$$  \n- $\\hat{R}$ 通过比较所有链上采样值的变异性和每个单独链上的变异性来解决一致性问题。  \n- combined 代表所有链的样本值的总变异。  \n- within 代表每个链内部样本值的变异。  \n\n理想情况下，$\\hat{R}$ ≈ 1，反映了平行链的稳定性。  \n- 相反，$\\hat{R}$ > 1表示链之间的变异不稳定，即组合链内的变异性大于单个链内的变异性。  \n- 虽然不存在黄金法则，但$\\hat{R}$比率大于1.05会给模拟的稳定性带来危险信号。  \n\n> 更详细的定义请参见 Vehtari 等人（2021）。如需了解有效样本量, R-hat等指标的的联系，请参阅 Vats 和 Knudson (2018)。"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"847C699BE6EB4BCB8192544FC369360F","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"对于示例1  \n\n![Image Name](https://cdn.kesci.com/upload/s2lg1juesc.png?imageView2/0/w/640/h/640)  \n- $\\hat{R}$ 值等于 1，这反映了四个并行链之间和链内部的变异性存在很好的一致性。  \n\n对于示例2  \n\n![Image Name](https://cdn.kesci.com/upload/s2lg5bgzbr.png?imageView2/0/w/640/h/640)  \n\n- $\\hat{R}$ = 5.35，这表明所有链值的方差总和是每个链内典型方差的 5 倍多。这远远超过了 1.05 的标准，充分证明假设平行链并没有产生一致的后验近似值。  \n- 因此 MCMC 的模拟近似是不稳定的。其后验分布的结果不可用于后续推断。  \n\n<div style=\"padding-bottom: 30px;\"></div>"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"4F5A102B243D4E5FB39330BD4E790108","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"## 代码演示 -- MCMC 诊断  \n\n使用 Stan 建模的模型默认使用 MCMC 方法进行参数估计。  \n\n且可以很方便的使用`bayesplots`包中 的`mcm_trace` 和 `mcmc_acf` 函数查看轨迹图和自相关图。"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"7DE8BBC3991844A1903C529F5381020A","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"首先，建立一个简单的`Beta-Binomial` 模型，完成MCMC采样 (以下术语可见课件上分术语表)  \n- 设置采样样本数 (iter) 为 2000  \n- 设置 MCMC 链 (chains) 的数量为 4，和上面的示例一样。  \n- 设置 MCMC 调试样本的数量 (warmup) 为0  \n    - 这里暂时理解 warmup 为提前进行 MCMC 采样若干次，然后丢弃掉这些采样。  \n    - 根据 MCMC 的性质，链中越早的采样越不稳定。因此，丢掉这些采样可以保证后续的采样更加稳定。  "},{"cell_type":"code","metadata":{"jupyter":{},"collapsed":false,"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"3FABF72795804BF18777A438429B5F73","notebookId":"68f9c684c011c75553f024e8","trusted":true},"source":"# 安装和加载包\noptions(repos = c(CRAN = \"https://mirrors.tuna.tsinghua.edu.cn/CRAN/\"))\nif (!requireNamespace('pacman', quietly = TRUE)) {\n    install.packages('pacman')\n}\npacman::p_load(\"tidyverse\",\"ggplot2\", \"dplyr\",\"gridExtra\",\"papaja\", \"patchwork\",\"bayesplot\",\"rstan\") # \"bayesrules\"\noptions(warn = -1)  # 抑制警告","outputs":[],"execution_count":1},{"cell_type":"code","metadata":{"jupyter":{},"collapsed":false,"scrolled":true,"tags":[],"slideshow":{"slide_type":"slide"},"id":"87A9632D45A24A5DAE2BA26DA2A1A78C","notebookId":"68f9c684c011c75553f024e8","trusted":true},"source":"# 准备数据\nstan_data <- list(N = 10,Y = 9)\n\n# 定义模型\nstan_model_code <- \"\ndata {\n  int<lower=0> N;       \n  int<lower=0, upper=N> Y; \n}\nparameters {\n  real<lower=0, upper=1> pi;\n}\nmodel {\n  pi ~ beta(2, 2);\n  Y ~ binomial(N, pi);\n}\n\"\n\n# 进行采样\nfit <- stan(\n  model_code = stan_model_code,      # 定义的模型或模型文件路径\n  data = stan_data,                  # 输入数据\n  chains = 4,                        # 马尔可夫链数量\n  iter = 2000,                       # 总迭代次数（每个链）\n  warmup = 0,                        # 热身迭代次数（不保存）\n  seed = 84735\n)","outputs":[{"output_type":"stream","name":"stdout","text":"\nSAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).\nChain 1: \nChain 1: Gradient evaluation took 5e-06 seconds\nChain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.05 seconds.\nChain 1: Adjust your expectations accordingly!\nChain 1: \nChain 1: \nChain 1: WARNING: No variance estimation is\nChain 1:          performed for num_warmup < 20\nChain 1: \nChain 1: Iteration:    1 / 2000 [  0%]  (Sampling)\nChain 1: Iteration:  200 / 2000 [ 10%]  (Sampling)\nChain 1: Iteration:  400 / 2000 [ 20%]  (Sampling)\nChain 1: Iteration:  600 / 2000 [ 30%]  (Sampling)\nChain 1: Iteration:  800 / 2000 [ 40%]  (Sampling)\nChain 1: Iteration: 1000 / 2000 [ 50%]  (Sampling)\nChain 1: Iteration: 1200 / 2000 [ 60%]  (Sampling)\nChain 1: Iteration: 1400 / 2000 [ 70%]  (Sampling)\nChain 1: Iteration: 1600 / 2000 [ 80%]  (Sampling)\nChain 1: Iteration: 1800 / 2000 [ 90%]  (Sampling)\nChain 1: Iteration: 2000 / 2000 [100%]  (Sampling)\nChain 1: \nChain 1:  Elapsed Time: 0 seconds (Warm-up)\nChain 1:                0.008 seconds (Sampling)\nChain 1:                0.008 seconds (Total)\nChain 1: \n\nSAMPLING FOR MODEL 'anon_model' NOW (CHAIN 2).\nChain 2: \nChain 2: Gradient evaluation took 2e-06 seconds\nChain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.02 seconds.\nChain 2: Adjust your expectations accordingly!\nChain 2: \nChain 2: \nChain 2: WARNING: No variance estimation is\nChain 2:          performed for num_warmup < 20\nChain 2: \nChain 2: Iteration:    1 / 2000 [  0%]  (Sampling)\nChain 2: Iteration:  200 / 2000 [ 10%]  (Sampling)\nChain 2: Iteration:  400 / 2000 [ 20%]  (Sampling)\nChain 2: Iteration:  600 / 2000 [ 30%]  (Sampling)\nChain 2: Iteration:  800 / 2000 [ 40%]  (Sampling)\nChain 2: Iteration: 1000 / 2000 [ 50%]  (Sampling)\nChain 2: Iteration: 1200 / 2000 [ 60%]  (Sampling)\nChain 2: Iteration: 1400 / 2000 [ 70%]  (Sampling)\nChain 2: Iteration: 1600 / 2000 [ 80%]  (Sampling)\nChain 2: Iteration: 1800 / 2000 [ 90%]  (Sampling)\nChain 2: Iteration: 2000 / 2000 [100%]  (Sampling)\nChain 2: \nChain 2:  Elapsed Time: 0 seconds (Warm-up)\nChain 2:                0.008 seconds (Sampling)\nChain 2:                0.008 seconds (Total)\nChain 2: \n\nSAMPLING FOR MODEL 'anon_model' NOW (CHAIN 3).\nChain 3: \nChain 3: Gradient evaluation took 2e-06 seconds\nChain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.02 seconds.\nChain 3: Adjust your expectations accordingly!\nChain 3: \nChain 3: \nChain 3: WARNING: No variance estimation is\nChain 3:          performed for num_warmup < 20\nChain 3: \nChain 3: Iteration:    1 / 2000 [  0%]  (Sampling)\nChain 3: Iteration:  200 / 2000 [ 10%]  (Sampling)\nChain 3: Iteration:  400 / 2000 [ 20%]  (Sampling)\nChain 3: Iteration:  600 / 2000 [ 30%]  (Sampling)\nChain 3: Iteration:  800 / 2000 [ 40%]  (Sampling)\nChain 3: Iteration: 1000 / 2000 [ 50%]  (Sampling)\nChain 3: Iteration: 1200 / 2000 [ 60%]  (Sampling)\nChain 3: Iteration: 1400 / 2000 [ 70%]  (Sampling)\nChain 3: Iteration: 1600 / 2000 [ 80%]  (Sampling)\nChain 3: Iteration: 1800 / 2000 [ 90%]  (Sampling)\nChain 3: Iteration: 2000 / 2000 [100%]  (Sampling)\nChain 3: \nChain 3:  Elapsed Time: 0 seconds (Warm-up)\nChain 3:                0.007 seconds (Sampling)\nChain 3:                0.007 seconds (Total)\nChain 3: \n\nSAMPLING FOR MODEL 'anon_model' NOW (CHAIN 4).\nChain 4: \nChain 4: Gradient evaluation took 2e-06 seconds\nChain 4: 1000 transitions using 10 leapfrog steps per transition would take 0.02 seconds.\nChain 4: Adjust your expectations accordingly!\nChain 4: \nChain 4: \nChain 4: WARNING: No variance estimation is\nChain 4:          performed for num_warmup < 20\nChain 4: \nChain 4: Iteration:    1 / 2000 [  0%]  (Sampling)\nChain 4: Iteration:  200 / 2000 [ 10%]  (Sampling)\nChain 4: Iteration:  400 / 2000 [ 20%]  (Sampling)\nChain 4: Iteration:  600 / 2000 [ 30%]  (Sampling)\nChain 4: Iteration:  800 / 2000 [ 40%]  (Sampling)\nChain 4: Iteration: 1000 / 2000 [ 50%]  (Sampling)\nChain 4: Iteration: 1200 / 2000 [ 60%]  (Sampling)\nChain 4: Iteration: 1400 / 2000 [ 70%]  (Sampling)\nChain 4: Iteration: 1600 / 2000 [ 80%]  (Sampling)\nChain 4: Iteration: 1800 / 2000 [ 90%]  (Sampling)\nChain 4: Iteration: 2000 / 2000 [100%]  (Sampling)\nChain 4: \nChain 4:  Elapsed Time: 0 seconds (Warm-up)\nChain 4:                0.008 seconds (Sampling)\nChain 4:                0.008 seconds (Total)\nChain 4: \n"}],"execution_count":2},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"D458DC5AEE70438593B1C7C12633E7CE","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"绘制 trace 图。  \n\n- 可以看到，每条链的后验分布都重合在一起。  \n- MCMC链的行为表现得类似噪音一样，这是一个好的表现。"},{"cell_type":"code","metadata":{"jupyter":{},"collapsed":false,"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"336E2B4D231A41D3B1CAC76B775D4691","notebookId":"68f9c684c011c75553f024e8","trusted":true},"source":"trace <- bayesplot::mcmc_trace(fit, pars = \"pi\") +  \n        papaja::theme_apa()\ntrace","outputs":[{"output_type":"display_data","data":{"text/plain":"plot without 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g3giBjSwxFACjZ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\n2uPsAD0wi+p2HuJjCMlbnNrV10hCrpDqA7Dz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tFlNjm+kSE4qvp8EUmpRUvYRxLFQJfK0QDj0lhLr2QgFUtI1TJmRCXuOWKGrWYtoBfcF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zFCbwpM4J0mGKAURCkkxx9xUfy23NX9yaRgyND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\nATTFmnWkrQ5cYjqj1bBa1qDXsByRfGeLzpotWFpq9pq9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fQn1jfgT2Kz5icp7gPqN1Duoz0zKuYH6OXVDwI2kbBV9146mTjBHPuY+oJsU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\ncmVhbQplbmRvYmoKc3RhcnR4cmVmCjU3MTMxCiUlRU9GCg==","text/html":"<img src=\"https://cdn.kesci.com/upload/rt/336E2B4D231A41D3B1CAC76B775D4691/t4krdtcwh3.svg\">"},"metadata":{"image/png":{"width":420,"height":420},"image/jpeg":{"width":420,"height":420},"image/svg+xml":{"width":420,"height":420,"isolated":true},"application/pdf":{"width":420,"height":420}}}],"execution_count":3},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"32421062F7BE4F3AB6287324605AFDD5","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"绘制自相关图  \n\n- 可以看到，随着间隔的下降，自相关也快速下降"},{"cell_type":"code","metadata":{"jupyter":{},"collapsed":false,"scrolled":true,"tags":[],"slideshow":{"slide_type":"slide"},"id":"70EE24A3E0FF4B29BC4998AEACF3427F","notebookId":"68f9c684c011c75553f024e8","trusted":true},"source":"options(repr.plot.width=8, repr.plot.height=16) \nauto_r <- bayesplot::mcmc_acf(fit, pars = \"pi\") + \n          papaja::theme_apa()\n\nauto_r","outputs":[{"output_type":"display_data","data":{"text/plain":"plot without 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src=\"https://cdn.kesci.com/upload/rt/70EE24A3E0FF4B29BC4998AEACF3427F/t4krdt9ylt.svg\">"},"metadata":{"image/png":{"width":480,"height":960},"image/jpeg":{"width":480,"height":960},"image/svg+xml":{"width":480,"height":960,"isolated":true},"application/pdf":{"width":480,"height":960}}}],"execution_count":4},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"568A2FB1028B48EFB8DCA41F41B95D85","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"使用 `bayesplot` 的 `rhat()` 和 `neff_ratio() `函数来获得对于 MCMC 链的 R-hat 和有效样本量比率 ESS Ratio指标。  \n\n- 可以看到 r-hat 接近1，表明链间和各链内的变异的一致性。  \n- 有效样本量比率  = 0.50，大于 0.1。"},{"cell_type":"code","metadata":{"jupyter":{},"collapsed":false,"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"BC8D8AF78C5C46B68C27C089FD2FAF29","notebookId":"68f9c684c011c75553f024e8","trusted":true},"source":"bayesplot::rhat(fit, pars = \"pi\")\nbayesplot::neff_ratio(fit, pars = \"pi\")","outputs":[{"output_type":"display_data","data":{"text/html":"1.00100399172714","text/markdown":"1.00100399172714","text/latex":"1.00100399172714","text/plain":"[1] 1.001004"},"metadata":{}},{"output_type":"display_data","data":{"text/html":"0.465807437977314","text/markdown":"0.465807437977314","text/latex":"0.465807437977314","text/plain":"[1] 0.4658074"},"metadata":{}}],"execution_count":5},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"D324DD0D18EB412888AEAC9AE6A41FAD","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"虽然示例的 trace 不存在强烈的自相关，还是可以看看如何通过 thin 参数来使得采样更薄，从而减少链的自相关。  \n\n- 设置 thin 为10，即每隔10保留一个参数，这导致样本数量只有之前的 1/10，即500  \n- 可以看到，自相关几乎不存在了。"},{"cell_type":"code","metadata":{"jupyter":{},"collapsed":false,"scrolled":true,"tags":[],"slideshow":{"slide_type":"slide"},"id":"DF9FB05C47714CB4B018677C042042AE","notebookId":"68f9c684c011c75553f024e8","trusted":true},"source":"# 进行采样\nthinned_fit <- rstan::stan(\n  model_code = stan_model_code,      # 定义的模型或模型文件路径\n  data = stan_data,                  # 输入数据\n  chains = 4,                        # 马尔可夫链数量\n  iter = 2000*2,                     # 总迭代次数（每个链）\n  thin = 10,                         # 缩减马尔可夫链\n  seed = 84735\n)\n","outputs":[{"output_type":"stream","name":"stdout","text":"\nSAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).\nChain 1: \nChain 1: Gradient evaluation took 3e-06 seconds\nChain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.03 seconds.\nChain 1: Adjust your expectations accordingly!\nChain 1: \nChain 1: \nChain 1: Iteration:    1 / 4000 [  0%]  (Warmup)\nChain 1: Iteration:  400 / 4000 [ 10%]  (Warmup)\nChain 1: Iteration:  800 / 4000 [ 20%]  (Warmup)\nChain 1: Iteration: 1200 / 4000 [ 30%]  (Warmup)\nChain 1: Iteration: 1600 / 4000 [ 40%]  (Warmup)\nChain 1: Iteration: 2000 / 4000 [ 50%]  (Warmup)\nChain 1: Iteration: 2001 / 4000 [ 50%]  (Sampling)\nChain 1: Iteration: 2400 / 4000 [ 60%]  (Sampling)\nChain 1: Iteration: 2800 / 4000 [ 70%]  (Sampling)\nChain 1: Iteration: 3200 / 4000 [ 80%]  (Sampling)\nChain 1: Iteration: 3600 / 4000 [ 90%]  (Sampling)\nChain 1: Iteration: 4000 / 4000 [100%]  (Sampling)\nChain 1: \nChain 1:  Elapsed Time: 0.007 seconds (Warm-up)\nChain 1:                0.007 seconds (Sampling)\nChain 1:                0.014 seconds (Total)\nChain 1: \n\nSAMPLING FOR MODEL 'anon_model' NOW (CHAIN 2).\nChain 2: \nChain 2: Gradient evaluation took 2e-06 seconds\nChain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.02 seconds.\nChain 2: Adjust your expectations accordingly!\nChain 2: \nChain 2: \nChain 2: Iteration:    1 / 4000 [  0%]  (Warmup)\nChain 2: Iteration:  400 / 4000 [ 10%]  (Warmup)\nChain 2: Iteration:  800 / 4000 [ 20%]  (Warmup)\nChain 2: Iteration: 1200 / 4000 [ 30%]  (Warmup)\nChain 2: Iteration: 1600 / 4000 [ 40%]  (Warmup)\nChain 2: Iteration: 2000 / 4000 [ 50%]  (Warmup)\nChain 2: Iteration: 2001 / 4000 [ 50%]  (Sampling)\nChain 2: Iteration: 2400 / 4000 [ 60%]  (Sampling)\nChain 2: Iteration: 2800 / 4000 [ 70%]  (Sampling)\nChain 2: Iteration: 3200 / 4000 [ 80%]  (Sampling)\nChain 2: Iteration: 3600 / 4000 [ 90%]  (Sampling)\nChain 2: Iteration: 4000 / 4000 [100%]  (Sampling)\nChain 2: \nChain 2:  Elapsed Time: 0.008 seconds (Warm-up)\nChain 2:                0.008 seconds (Sampling)\nChain 2:                0.016 seconds (Total)\nChain 2: \n\nSAMPLING FOR MODEL 'anon_model' NOW (CHAIN 3).\nChain 3: \nChain 3: Gradient evaluation took 2e-06 seconds\nChain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.02 seconds.\nChain 3: Adjust your expectations accordingly!\nChain 3: \nChain 3: \nChain 3: Iteration:    1 / 4000 [  0%]  (Warmup)\nChain 3: Iteration:  400 / 4000 [ 10%]  (Warmup)\nChain 3: Iteration:  800 / 4000 [ 20%]  (Warmup)\nChain 3: Iteration: 1200 / 4000 [ 30%]  (Warmup)\nChain 3: Iteration: 1600 / 4000 [ 40%]  (Warmup)\nChain 3: Iteration: 2000 / 4000 [ 50%]  (Warmup)\nChain 3: Iteration: 2001 / 4000 [ 50%]  (Sampling)\nChain 3: Iteration: 2400 / 4000 [ 60%]  (Sampling)\nChain 3: Iteration: 2800 / 4000 [ 70%]  (Sampling)\nChain 3: Iteration: 3200 / 4000 [ 80%]  (Sampling)\nChain 3: Iteration: 3600 / 4000 [ 90%]  (Sampling)\nChain 3: Iteration: 4000 / 4000 [100%]  (Sampling)\nChain 3: \nChain 3:  Elapsed Time: 0.007 seconds (Warm-up)\nChain 3:                0.006 seconds (Sampling)\nChain 3:                0.013 seconds (Total)\nChain 3: \n\nSAMPLING FOR MODEL 'anon_model' NOW (CHAIN 4).\nChain 4: \nChain 4: Gradient evaluation took 2e-06 seconds\nChain 4: 1000 transitions using 10 leapfrog steps per transition would take 0.02 seconds.\nChain 4: Adjust your expectations accordingly!\nChain 4: \nChain 4: \nChain 4: Iteration:    1 / 4000 [  0%]  (Warmup)\nChain 4: Iteration:  400 / 4000 [ 10%]  (Warmup)\nChain 4: Iteration:  800 / 4000 [ 20%]  (Warmup)\nChain 4: Iteration: 1200 / 4000 [ 30%]  (Warmup)\nChain 4: Iteration: 1600 / 4000 [ 40%]  (Warmup)\nChain 4: Iteration: 2000 / 4000 [ 50%]  (Warmup)\nChain 4: Iteration: 2001 / 4000 [ 50%]  (Sampling)\nChain 4: Iteration: 2400 / 4000 [ 60%]  (Sampling)\nChain 4: Iteration: 2800 / 4000 [ 70%]  (Sampling)\nChain 4: Iteration: 3200 / 4000 [ 80%]  (Sampling)\nChain 4: Iteration: 3600 / 4000 [ 90%]  (Sampling)\nChain 4: Iteration: 4000 / 4000 [100%]  (Sampling)\nChain 4: \nChain 4:  Elapsed Time: 0.007 seconds (Warm-up)\nChain 4:                0.007 seconds (Sampling)\nChain 4:                0.014 seconds (Total)\nChain 4: \n"}],"execution_count":6},{"cell_type":"code","metadata":{"jupyter":{},"collapsed":false,"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"BE3A961954CC47D2A5E8BB85AD10B74B","notebookId":"68f9c684c011c75553f024e8","trusted":true},"source":"options(repr.plot.width=10, repr.plot.height=5) \nthinned_trace <- bayesplot::mcmc_trace(thinned_fit, pars = \"pi\") +  \n        papaja::theme_apa()\n\nthinned_trace","outputs":[{"output_type":"display_data","data":{"text/plain":"plot without 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src=\"https://cdn.kesci.com/upload/rt/BE3A961954CC47D2A5E8BB85AD10B74B/t4krdu2d5d.svg\">"},"metadata":{"image/png":{"width":600,"height":300},"image/jpeg":{"width":600,"height":300},"image/svg+xml":{"width":600,"height":300,"isolated":true},"application/pdf":{"width":600,"height":300}}}],"execution_count":7},{"cell_type":"code","metadata":{"jupyter":{},"collapsed":false,"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"C133F12DAF6F4A2D995CA116F41EFC73","notebookId":"68f9c684c011c75553f024e8","trusted":true},"source":"options(repr.plot.width=12, repr.plot.height=14) \np1 <- bayesplot::mcmc_acf_bar(fit, pars = \"pi\", lags = 10) + \n      papaja::theme_apa() + \n      ggplot2::ggtitle('Full Trace')\n\np2 <- bayesplot::mcmc_acf_bar(thinned_fit, pars = \"pi\", lags = 10) + \n      papaja::theme_apa() +\n      ggplot2::ggtitle('Thinned Trace')\n\np1 + p2","outputs":[{"output_type":"display_data","data":{"text/plain":"plot without title","application/pdf":"JVBERi0xLjcKJbXtrvsKNCAwIG9iago8PCAvTGVuZ3RoIDUgMCBSCiAgIC9GaWx0ZXIgL0Zs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\ncpwQtphdjdxqiENK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7Zrf+Bf/vtpZNfur2v569/3843/QL4xG5WCmVuZHN0cmVh\nbQplbmRvYmoKMTggMCBvYmoKICAgMjc3CmVuZG9iagoxOSAwIG9iago8PCAvVHlwZSAvWFJl\nZgogICAvTGVuZ3RoIDgxCiAgIC9GaWx0ZXIgL0ZsYXRlRGVjb2RlCiAgIC9TaXplIDIwCiAg\nIC9XIFsxIDIgMl0KICAgL1Jvb3QgMTcgMCBSCiAgIC9JbmZvIDE2IDAgUgo+PgpzdHJlYW0K\neJxjYGD4/5+JgYuBAUQwMXKfZGBgZOAHEtybQGIcQJbaCyDBYwYidgAJzUAQ6yyQUP4FJFSE\ngITqHhBxGWIKI4hgZtS6DRTT+sTAAABycAyhCmVuZHN0cmVhbQplbmRvYmoKc3RhcnR4cmVm\nCjEwOTk0CiUlRU9GCg==","text/html":"<img src=\"https://cdn.kesci.com/upload/rt/C133F12DAF6F4A2D995CA116F41EFC73/t4krdwexwp.svg\">"},"metadata":{"image/png":{"width":720,"height":840},"image/jpeg":{"width":720,"height":840},"image/svg+xml":{"width":720,"height":840,"isolated":true},"application/pdf":{"width":720,"height":840}}}],"execution_count":8},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"1A82B2C6688D47FBA3A6E27DF5C7E66C","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"## 利用后验分布进行推断: 参数估计(Posterior estimation)  \n\n当确定 MCMC 采样结果可以相信之后，就可以将其**当作后验分布**进行统计推断。  \n\n再回顾后验分布的含义和意义。  \n"},{"cell_type":"code","metadata":{"id":"BD91C2B81162433391A768D63C880BD2","notebookId":"68f9c684c011c75553f024e8","jupyter":{},"collapsed":false,"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"trusted":true},"source":"# 使用bayesrules包中的 plot_beta_binomial（）函数可以绘制先验分布、似然分布和后验分布的 PDF 图\n# 可以使用 `install.packages(\"bayesrules\")`安装该包，但是该包涉及到大量的依赖，安装比较费时\n# 所以我们直接把这个函数复制过来（https://github.com/bayes-rules/bayesrules/blob/master/R/plot_beta_binomial.R）：\nlibrary(tidyverse)\nplot_beta_binomial <- function (alpha,\n                                beta,\n                                y = NULL,\n                                n = NULL,\n                                prior = TRUE,\n                                likelihood = TRUE,\n                                posterior = TRUE){\n  if (is.null(y) | is.null(n))\n    warning(\"To visualize the posterior,\n            specify data y and n\")\n\n  g <- ggplot(data = data.frame(x = c(0, 1)), aes(x)) +\n    labs(x = expression(pi),\n         y = \"density\") +\n    scale_fill_manual(\"\",\n                      values = c(prior = \"#f0e442\",\n                                 `(scaled) likelihood` = \"#0071b2\",\n                                 posterior = \"#009e74\"),\n                      breaks = c(\"prior\",\n                                 \"(scaled) likelihood\",\n                                 \"posterior\"))\n  \n  if (prior == TRUE) {\n    g <- g +\n      stat_function(fun = dbeta,\n                           args = list(shape1 = alpha,\n                                       shape2 = beta)) +\n      stat_function(fun = dbeta,\n                    args = list(shape1 = alpha,\n                                shape2 = beta),\n                    geom = \"area\",\n                    alpha = 0.5,\n                    aes(fill = \"prior\"))\n    }\n\n  if (!is.null(y) & !is.null(n)) {\n    alpha_post <- alpha + y\n    beta_post <- beta + n - y\n    y_data <- y\n    like_scaled <- function(x) {\n      like_fun <- function(x) {\n        dbinom(x = y_data, size = n, prob = x)\n      }\n      scale_c <- integrate(like_fun, lower = 0, upper = 1)[[1]]\n      like_fun(x)/scale_c\n    }\n  }\n  if (!is.null(y) & !is.null(n) & (likelihood != FALSE)) {\n    g <- g +\n      stat_function(fun = like_scaled) +\n      stat_function(fun = like_scaled,\n                    geom = \"area\",\n                    alpha = 0.5,\n                    aes(fill = \"(scaled) likelihood\"))\n  }\n  if (!is.null(y) & !is.null(n) & posterior == TRUE) {\n    g <- g +\n      stat_function(fun = dbeta,\n                    args = list(shape1 = alpha_post,\n                                shape2 = beta_post)) +\n      stat_function(fun = dbeta,\n                    args = list(shape1 = alpha_post,\n                                shape2 = beta_post),\n                    geom = \"area\", alpha = 0.5,\n                    aes(fill = \"posterior\"))\n  }\n  g\n} # end of function\n","outputs":[],"execution_count":9},{"cell_type":"code","metadata":{"jupyter":{},"collapsed":false,"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"72171CDDA7EC4A2688390054DA526DA3","notebookId":"68f9c684c011c75553f024e8","trusted":true},"source":"# 使用bayesrules包中的 plot_beta_binomial（）函数可以绘制先验分布、似然分布和后验分布的 PDF 图\nfig <- plot_beta_binomial(alpha = 2, beta = 2, y = 9, n = 10)  +\n       papaja::theme_apa() +\n       scale_y_continuous(expand = c(0,0)) + \n       theme(legend.text = element_text(size = 16))\nprint(fig)","outputs":[{"output_type":"display_data","data":{"text/plain":"plot without 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KizDV+T6SuSJWI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\nOpdzTcRsvF6KPWQMpWZqlf4GNV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src=\"https://cdn.kesci.com/upload/rt/72171CDDA7EC4A2688390054DA526DA3/t4krdxaslx.svg\">"},"metadata":{"image/png":{"width":720,"height":840},"image/jpeg":{"width":720,"height":840},"image/svg+xml":{"width":720,"height":840,"isolated":true},"application/pdf":{"width":720,"height":840}}}],"execution_count":10},{"cell_type":"markdown","metadata":{"id":"F81C6505446C4F7A90A0112DF946AA6D","notebookId":"68f9c684c011c75553f024e8","runtime":{"status":"default","execution_status":null,"is_visible":false},"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"}},"source":"## 利用后验分布进行推断: 参数估计(Posterior estimation)  \n\n**在贝叶斯的框架下，我们可以把后验分布看作是对成功次数$\\pi$的一种估计，即当前数据更可能在哪一种$\\pi$下出现。**  \n\n> 🤔当你看到这个后验分布时，你觉得它描述了关于$\\pi$怎样的一种信息呢？  \n> * a. 参与者正确判断的概率大概在80%  \n> * b. 参与者正确判断的概率比例最有可能是80%，但这个比例也可能是60%到99%中的某一个。  "},{"cell_type":"markdown","metadata":{"id":"7BA9836D62084CDC849DABF8D4089EFC","jupyter":{},"notebookId":"68f9c684c011c75553f024e8","runtime":{"status":"default","execution_status":null,"is_visible":false},"scrolled":false,"slideshow":{"slide_type":"slide"},"tags":[]},"source":"**选项b是更符合贝叶斯取向的回答**。 为什么？ ➡  \n\n* 我们来考虑另一种调查情况下得出的后验分布：  \n\n* 我们对$\\pi$的先验仍然为$\\pi \\sim Beta(2, 2)$，但现在我们的试验总次数为118次，成功次数为99次.  \n\n$$  \n\\pi \\sim Beta(2, 2)  \n$$  \n\n$$  \nY | \\pi  \\sim Bin(118, \\pi)  \n$$  \n\n$$  \n\\pi | (y = 99) \\sim \\text{Beta}(\\alpha + y, \\beta + n - y)  \n$$  \n\n* 这种情况下，$\\pi$的后验分布满足$\\pi | (Y = 99) \\sim Beta(101, 21)$"},{"cell_type":"markdown","metadata":{"id":"AFF3361D18E743B0B715E0FB49DA7470","jupyter":{},"notebookId":"68f9c684c011c75553f024e8","runtime":{"status":"default","execution_status":null,"is_visible":false},"scrolled":false,"slideshow":{"slide_type":"slide"},"tags":[]},"source":"下图展示了这两种情况下的后验分布  \n\n> 黑线表示众数，即在后验分布中最可能出现的值  \n\n![Image Name](https://cdn.kesci.com/upload/sm3w84q11g.png?imageView2/0/w/700)"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"A93B962E6A204165BED662DD91D893A9","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"\n* 在这两种分布下，$\\pi$最可能的取值都在83-84%左右  \n\n* 如果只关注众数，两种截然不同的情况却会导致相似的结论❌  \n\n* 但很明显，我们可以看到两图中$\\pi$的分布范围是不同的  \n\n\n**要同时看到$\\pi$的集中趋势和离散趋势，可使用 *94%的可信区间(credible intervals, CrI)* 来描述$\\pi$**"},{"cell_type":"markdown","metadata":{"id":"0AA030D9188E46768478B5CD6F65A0E5","jupyter":{},"notebookId":"68f9c684c011c75553f024e8","runtime":{"status":"default","execution_status":null,"is_visible":false},"scrolled":false,"slideshow":{"slide_type":"slide"},"tags":[]},"source":"### 可信区间(credible intervals)  \n\n* 后验分布的3%百分位数和97%百分位数包含的区域共同组成了94%的后验可信区间  \n\n* 可信区间表示了参数出现在这个区域的概率  \n    * 比如，Beta(101, 21) 94%的可信区间为(0.756, 0.889)。这意味着我们可以说，在随机点运动任务中参与者成功的真实概率出现在(0.756, 0.889)的可能性是95%  \n    * 一种理解是，有95%的人在随机点运动任务中参与者成功概率很高，大概为 0.756 到 0.889。  \n"},{"cell_type":"code","metadata":{"jupyter":{},"collapsed":false,"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"8711CA76C64848AC89FDA58DEFB6C646","notebookId":"68f9c684c011c75553f024e8","trusted":true},"source":"options(repr.plot.width=15, repr.plot.height=5) \n# 创建 Beta 分布的数据\nx_values <- seq(0, 1, length.out = 100)\ny_values1 <- dbeta(x_values, shape1 = 11, shape2 = 3)\ny_values2 <- dbeta(x_values, shape1 = 101, shape2 = 21)\n\n# 计算 95% 可信区间（CI）\n  lower_bound1_94 <- qbeta(0.03, shape1 = 11, shape2 = 3)  \n  upper_bound1_94 <- qbeta(0.97, shape1 = 11, shape2 = 3)  \n  lower_bound2_94 <- qbeta(0.03, shape1 = 101, shape2 = 21)  \n  upper_bound2_94 <- qbeta(0.97, shape1 = 101, shape2 = 21)  \n\n# 绘制 Beta 分布的 PDF\np1 <- ggplot2::ggplot(data.frame(x = x_values, y = y_values1),aes(x = x, y = y)) +\n  geom_line(col = 'black', size = 1) +\n  geom_area(data = subset(data.frame(x = x_values, y = y_values1),\n                           x >= lower_bound1_94 & x <= upper_bound1_94), \n            aes(x = x, y = y), fill = \"lightblue\", alpha = 0.5) +\n  labs( x = \" \", y = NULL, title = \"Continuous Prior\\n(Beta(alpha = 11, beta = 3))\")  +\n  scale_y_continuous(expand = c(0,0),limit = c(0,12)) + \n  papaja::theme_apa()\n\np2 <- ggplot2::ggplot(data.frame(x = x_values, y = y_values2),aes(x = x, y = y)) +\n  geom_line(col = 'black', size = 0.7) +\n  geom_area(data = subset(data.frame(x = x_values, y = y_values2),\n                           x >= lower_bound2_94 & x <= upper_bound2_94), \n            aes(x = x, y = y), fill = \"lightblue\", alpha = 0.5) +\n  labs( x = \" \", y = NULL, title = \"Continuous Prior\\n(Beta(alpha = 101, beta = 21))\")  +\n  scale_y_continuous(expand = c(0,0),limit = c(0,12)) + \n  papaja::theme_apa()\n\n#并排显示\np1 + p2","outputs":[{"output_type":"display_data","data":{"text/plain":"plot without 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frwSxkE/HgcFdH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\nJVgKWwplbmRzdHJlYW0KZW5kb2JqCnN0YXJ0eHJlZgoxMDQzMgolJUVPRgo=","text/html":"<img src=\"https://cdn.kesci.com/upload/rt/8711CA76C64848AC89FDA58DEFB6C646/t4krdxz0te.svg\">"},"metadata":{"image/png":{"width":900,"height":300},"image/jpeg":{"width":900,"height":300},"image/svg+xml":{"width":900,"height":300,"isolated":true},"application/pdf":{"width":900,"height":300}}}],"execution_count":11},{"cell_type":"markdown","metadata":{"id":"769D68E3327D445A9421DC5BE2701D01","jupyter":{},"notebookId":"68f9c684c011c75553f024e8","runtime":{"status":"default","execution_status":null,"is_visible":false},"scrolled":false,"slideshow":{"slide_type":"slide"},"tags":[]},"source":"**在报告可信区间时，95%是最常见的选项，但并不是唯一的选项。**  \n\n* 我们也可以创建其他可信区间，比如Beta(101, 21)的分布之下：  \n\t* 50% 可信区间：由25%百分位数和75%百分位数包含的区域  \n\t* 99%可信区间：由0.5%百分位数和99.5%百分位数包含的区域  \n\n> 在95%CI的图中，由于后验分布是一个偏态分布，可以看到相较于前半部分，后半部分包含的值对应的概率更低。  \n> 当后验分布是偏态分布时，有时我们不以中位数为中心构建95%的可信区间，而是基于最高后验概率密度(众数)。  \n"},{"cell_type":"code","metadata":{"jupyter":{},"collapsed":false,"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"74E459B7E02B4D44953D7712856B5DC6","notebookId":"68f9c684c011c75553f024e8","trusted":true},"source":"# 创建 Beta 分布的数据\nx_values <- seq(0, 1, length.out = 100)\ny_values <- dbeta(x_values, shape1 = 101, shape2 = 21)\n\n# 计算三种可信区间（CI）\n  lower_bound_94 <- qbeta(0.03, shape1 = 101, shape2 = 21)  \n  upper_bound_94 <- qbeta(0.97, shape1 = 101, shape2 = 21)  \n  lower_bound_50 <- qbeta(0.25, shape1 = 101, shape2 = 21)  \n  upper_bound_50 <- qbeta(0.75, shape1 = 101, shape2 = 21)  \n  lower_bound_99 <- qbeta(0.005, shape1 = 101, shape2 = 21)  \n  upper_bound_99 <- qbeta(0.995, shape1 = 101, shape2 = 21)  \n\n# 绘制 Beta 分布的 PDF\np1 <- ggplot2::ggplot(data.frame(x = x_values, y = y_values),aes(x = x, y = y)) +\n  geom_line(col = 'black', size = 1) +\n  geom_area(data = subset(data.frame(x = x_values, y = y_values),\n                           x >= lower_bound_50 & x <= upper_bound_50), \n            aes(x = x, y = y), fill = \"#bdd7e7\", alpha = 0.5) +\n  labs( x = \" \", y = NULL, title = \"50% CI for Beta(101, 21)\")  +\n  scale_y_continuous(expand = c(0,0),limit = c(0,12)) + \n  xlim(0.5,1) +\n  papaja::theme_apa()\n\n\np2 <- ggplot2::ggplot(data.frame(x = x_values, y = y_values),aes(x = x, y = y)) +\n  geom_line(col = 'black', size = 0.7) +\n  geom_area(data = subset(data.frame(x = x_values, y = y_values),\n                           x >= lower_bound_94 & x <= upper_bound_94), \n            aes(x = x, y = y), fill = \"#6baed6\", alpha = 0.5) +\n  labs( x = \" \", y = NULL, title = \"94% CI for Beta(101, 21)\")  +\n  scale_y_continuous(expand = c(0,0),limit = c(0,12)) + \n  xlim(0.5,1) +\n  papaja::theme_apa()\n\np3 <- ggplot2::ggplot(data.frame(x = x_values, y = y_values),aes(x = x, y = y)) +\n  geom_line(col = 'black', size = 0.7) +\n  geom_area(data = subset(data.frame(x = x_values, y = y_values),\n                           x >= lower_bound_99 & x <= upper_bound_99), \n            aes(x = x, y = y), fill = \"#3182bd\", alpha = 0.5) +\n  labs( x = \" \", y = NULL, title = \"99% CI for Beta(101, 21)\")  +\n  scale_y_continuous(expand = c(0,0)) + \n  xlim(0.5,1) +\n  papaja::theme_apa()\n\n#并排显示\noptions(repr.plot.width=18, repr.plot.height=5) \np1 + p2 + p3\n","outputs":[{"output_type":"display_data","data":{"text/plain":"plot without 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src=\"https://cdn.kesci.com/upload/rt/74E459B7E02B4D44953D7712856B5DC6/t4krdyf79c.svg\">"},"metadata":{"image/png":{"width":1080,"height":300},"image/jpeg":{"width":1080,"height":300},"image/svg+xml":{"width":1080,"height":300,"isolated":true},"application/pdf":{"width":1080,"height":300}}}],"execution_count":12},{"cell_type":"code","metadata":{"id":"45878588140E447BBC4B9DEB1069B54A","notebookId":"68f9c684c011c75553f024e8","jupyter":{},"collapsed":false,"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"trusted":true},"source":"### 练习: 使用stan模型来替代以上使用共轭求解的方法获得后验，并画出50%、84%、99%的可信区间","outputs":[],"execution_count":13},{"cell_type":"markdown","metadata":{"id":"7B928A28983542E08D5D1EDFDD4836CC","jupyter":{},"notebookId":"68f9c684c011c75553f024e8","runtime":{"status":"default","execution_status":null,"is_visible":false},"scrolled":false,"slideshow":{"slide_type":"slide"},"tags":[]},"source":"\n### 最大后验概率密度区间(the highest density interval, HDI)  \n\n最大概率密度区间HDI与中心区间是非常接近的。但是当后验分布为非正态分布时，两者存在区别，比如：  \n\n\n![Image Name](https://cdn.kesci.com/upload/image/rjdef48vqj.png?imageView2/0/w/640/h/640)  \n\n- 此时，后验中心区间从中位数的两侧开始展开。  \n- 而最大概率密度区间HDI从y轴最高的两处开始展开。  "},{"cell_type":"markdown","metadata":{"id":"EA24215D088A456F962CFC107E17EB6E","jupyter":{},"notebookId":"68f9c684c011c75553f024e8","runtime":{"status":"default","execution_status":null,"is_visible":false},"scrolled":false,"slideshow":{"slide_type":"slide"},"tags":[]},"source":"直觉理解：想象有一根水平的线从后验概率密度最高点往下滑，如果是95%HDI，则意味着黄色部分的面积占曲线下面积的95%  \n\n![Image Name](https://cdn.kesci.com/upload/s37y41n2yf.png?imageView2/0/w/640/h/640)  \n\n> source: https://mathematica.stackexchange.com/questions/173282/computing-credible-region-highest-posterior-density-from-empirical-distributio  \n\n<div style=\"padding-bottom: 30px;\"></div>"},{"cell_type":"markdown","metadata":{"id":"00CE3389D019435BB23F2BE2A023DCAB","jupyter":{},"notebookId":"68f9c684c011c75553f024e8","runtime":{"status":"default","execution_status":null,"is_visible":false},"scrolled":false,"slideshow":{"slide_type":"slide"},"tags":[]},"source":"### 95% CI vs 95%HDI  \n\n![Image Name](https://cdn.kesci.com/upload/s36vflrg1f.png?imageView2/0/w/960/h/960) "},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"B63A31E56C9A4BC092E72B68ADF3DF14","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"### 总结  \n通过概率编程语言中的MCMC samplers完成对后验分布的近似后，我们可以开始进行推断统计。  \n\n但进行推断之前，首先要检查 MCMC 的质量。  \n- 可视化：直观地检查多个并行链的轨迹图和密度图，以确定模拟的稳定性和混合性。  \n- 数学指标：用有效样本量比、自相关性和 $\\hat{R}$等数值诊断来补充这些视觉诊断。"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"FF080C3026814F6FB6BE153135D25BAA","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"在对后验进行推断中，要注意非正态分布后验的平均值、中位数和众数并不相等。  \n\n- 其中众数为分布最高点对应的参数值  \n- 中位数左右两侧分布的面积各占50%  \n- 平均值的位置接近中位数，并且它容易分布形态的影响  \n\n![Image Name](https://cdn.kesci.com/upload/image/rjddhc2wo.png?imageView2/0/w/500/h/500)  \n\n> source: book--A Student’s Guide to Bayesian Statistics  \n\n<div style=\"padding-bottom: 20px;\"></div>"},{"cell_type":"markdown","metadata":{"id":"BD0CCDDA0E3D4389895E873F1C37321B","jupyter":{},"notebookId":"68f9c684c011c75553f024e8","runtime":{"status":"default","execution_status":null,"is_visible":false},"scrolled":false,"slideshow":{"slide_type":"slide"},"tags":[]},"source":"## 利用后验分布进行推断: 假设检验  \n\n后验推断能告诉我们 “反应的正确率为多少”的估计范围。  \n\n但在传统的研究思路中，通常关心的问题是“正确率是否大于随机水平50%”（即虚无假设显著性检验）。在贝叶斯框架下应该完成这类推断？  \n\n* HDI + ROPE (最高密度区间 + 预设效应范围): 在贝叶斯分析中，HDI（Highest Density Interval）代表后验分布中概率最高的区间，而ROPE（Region of Practical Equivalence）则是一个预设的效果范围，用于判断效果是否具有实际意义。当HDI与ROPE重叠时，说明结果在实际应用中可能“不显著”；反之，则可能具有实际意义。  \n\n> Heck, D. W., Boehm, U., Böing-Messing, F., Bürkner, P.-C., Derks, K., Dienes, Z., Fu, Q., Gu, X., Karimova, D., Kiers, H. A. L., Klugkist, I., Kuiper, R. M., Lee, M. D., Leenders, R., Leplaa, H. J., Linde, M., Ly, A., Meijerink-Bosman, M., Moerbeek, M., … Hoijtink, H. (2023). A review of applications of the bayes factor in psychological research. Psychological Methods, 28(3), 558–579. https://doi.org/10.1037/met0000454  \n\n* Bayes Factor (贝叶斯因子): 贝叶斯因子是用于比较两种假设（通常是虚无假设与备择假设）相对支持程度的比率。它衡量的是数据在这两种假设下的解释能力。一个大于1的贝叶斯因子表示数据支持备择假设，而小于1则表示支持虚无假设。贝叶斯因子提供了一种量化不确定性的方法，能够帮助我们更好地理解结果的可靠性。  \n\n> Kruschke, J. K. (2018). Rejecting or accepting parameter values in bayesian estimation. Advances in Methods and Practices in Psychological Science, 1(2), 270–280. https://doi.org/10.1177/2515245918771304"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"9E85EE44DDA94598AD41FF6C18FB4170","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"\n>🤔假设我们在一篇报道中看到这样一种描述：在随机点运动任务中，成功次数占比为80%。  \n> 如何根据这个数据结果判断改成功率是否大于随机判断水平 50% 呢？  \n\n\n* 我们同时考虑两种可能性：成功次数 $\\pi$ 的比例是否与 50% 相同，此时备择假设和虚无假设可以写成：  \n$$  \n\\begin{split}  \nH_0: & \\; \\; \\pi = 0.5 \\\\  \nH_a: & \\; \\; \\pi \\ne 0.5 \\\\  \n\\end{split}  \n$$  "},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"30D25AE6F9604316AB13B58BE7879AE2","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"\n### 1. 通过 HDI 进行检验  \n\n沿用之前的例子，已经知道先验分布为 $Beta(2, 2)$； 后验分布为 $Beta(101, 21)$。  \n\n为了检验 $\\pi$ 是否显著不同于 0.5，我们可以计算 **后验分布的 HDI**（Highest Density Interval），这是一种包含后验分布中最高概率密度的区间。通常 HDI 表示一个区间，比如 95% HDI，意味着该区间包含了后验分布中的 95% 的概率质量。"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"A5EC1A8390884A3D9FBA160CB73636F0","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"**计算 HDI**  \n\nHDI 是后验分布中包含最高密度的区间，我们通常计算 95% HDI，这意味着该区间包含后验分布中 95% 的概率质量。HDI 的定义要求该区间 $[\\pi_\\text{low}, \\pi_\\text{high}]$ 满足以下条件：  \n\n1. 对于任何 $\\pi$ 在区间 $[\\pi_\\text{low}, \\pi_\\text{high}]$ 外的点，它的密度都小于区间内的点的密度，即：  \n\n$$  \nf(\\pi_\\text{low} | \\alpha, \\beta) = f(\\pi_\\text{high} | \\alpha, \\beta)  \n$$  \n\n2. 该区间内的概率质量为 95%：  \n\n$$  \n\\int_{\\pi_\\text{low}}^{\\pi_\\text{high}} f(\\pi | \\alpha, \\beta) d\\pi = 0.95  \n$$  \n"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"001021459B6C47DA9B79313E32B4409F","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"**结果解释**  \n\n经过计算，得到的 95% HDI 区间为 $[0.76, 0.89]$。接下来，我们要验证这个区间是否包含 0.5：  \n\n- 该区间为 $[0.76, 0.89]$。  \n- 0.5 不在这个区间内。  \n\n因此，95% HDI 区间不包含 0.5，表明在 95% 的可信水平下，成功率 $\\pi$ 与 0.5 存在差异。基于这一结果，我们可以拒绝虚无假设 $H_0: \\pi = 0.5$，认为成功率 $\\pi$ 显著不同于 0.5。"},{"cell_type":"code","metadata":{"jupyter":{},"collapsed":false,"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"6A9F322BD84142EA8E1B9CA6B598017D","notebookId":"68f9c684c011c75553f024e8","trusted":true},"source":"#可以使用bayestestR包中的hdi函数计算HDI的上界和下界\nlibrary(bayestestR)\n# 创建 Beta 分布的数据\nx_values <- seq(0, 1, length.out = 10000)\ny_values <- dbeta(x_values, shape1 = 101, shape2 = 21)\n\n# 创建数据框\nbeta_df <- data.frame(x = x_values, y = y_values)\n\n# 计算 HDI\nhdi_result <- hdi(rbeta(100000, shape1 = 101, shape2 = 21), pd = 0.95)\n\n# 提取 HDI 的下界和上界\nhdi_lower <- hdi_result$CI_low\nhdi_upper <- hdi_result$CI_high\n\n\n# 绘图\np <- ggplot(beta_df, aes(x = x, y = y)) +\n  geom_line(color = 'black', size = 1) +  # 绘制 PDF\n  geom_area(data = beta_df[(beta_df$x >= hdi_lower) & (beta_df$x <= hdi_upper), ],\n            aes(x = x, y = y), fill = 'lightblue', alpha = 0.5) +  # 填充 HDI 区域\n  geom_vline(xintercept = 0.5, linetype = \"dashed\", color = \"red\") +  # pi=0.5\n  labs(x = \"x\", y = \"Density\", title = \"Beta Distribution PDF and 95% HDI\") +\n  xlim(0.4, 1) +\n  papaja::theme_apa()\n\n# 输出 HDI 结果\ncat(\"95% HDI 下界:\", hdi_lower, \"\\n\")\ncat(\"95% HDI 上界:\", hdi_upper, \"\\n\")\n# 显示图形\noptions(repr.plot.width=8, repr.plot.height=6) \nprint(p)\n","outputs":[{"output_type":"stream","name":"stderr","text":"\nAttaching package: ‘bayestestR’\n\n\nThe following object is masked from ‘package:papaja’:\n\n    ci\n\n\n"},{"output_type":"stream","name":"stdout","text":"95% HDI 下界: 0.7596789 \n95% HDI 上界: 0.8921641 \n"},{"output_type":"display_data","data":{"text/plain":"plot without 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HW2sw3WEF5cHDPqIiA/LACCYAPkuzzYHYhsQQkT3uwGm6lKf61MIB+Tc\nt8sI4T0gB4AQ+SnakBgPgN0tJeO9iBZh4YSNDSsg41Cd6PbJqkKK2H3pBowoYgNn5yGU/3vr\nnirCRWn3tpMOgrTfj0FHsBG2B9tgWjUiQgq38VYWwS8Ke2N3iLuL28dgIwxADtiQ185tTD8Z\nRc+x1Zk3jFsdl/ZDJSzYDYIQYvNVPgC3Mjf3Zz1Pg8D9uxUTEFkTqUDYQ675NQb+sBetAKvT\nCLiFrVnZqTLY5rZ5VCrYpQ10ueND2Yzah18mRwmweW///ACoHjMESp4LgnQxYyGHOf9gKoRc\nkgOgFkyykEM2BiWe6SPdAALPtJSQRQDaW+ktltTBgKgyZ5JAsDMBRyR4yf2JCR8Ew4lkI682\nD8HtRMrUzuvZHLnlpuuY1Arma9l+HoQNKGR+yWqMyb8xp8zqNbn8AbGbewh0CYlpU1kbQINn\nL7n9BbKUTOdbOKlfzMCTlz0mZR3JfbJ9RzIXg+9IHFQKIcmZF5G2aBM1FpUaxP7ZWDRozH9U\nAiUQAjNIm3RcH2QXEs8wTrdl1qYFaHM7CcQpnUSrMvsz5YVBLIAcZzOKCOIg0JAoIdF0aZNl\nINnkzDLMWB2JEKMH6Xl2ROiZ57oMcCySYx2NHPITQY5FoShM5Jwwx9bu4gGIfoDY75s310PI\n1Z1aF01N8yCYJi8hRCWU4ptfgzbXuTJoeAjuFIjDwPPmggS5wniWHggqD+S1CshReRBvDE1F\ncpGGLFmYhcpcZ9ceyyW0FpEXk7NwzQ/iKP4hnRnPFs2pNTaypJ9xF2b3zJsWwrRV8SdZu/M3\nqu4sOV047xrEjp++j4Fuzs9CMIUVz/8hiOxXCL9P1hjXrGM45xMkYA4qkyiKEbQYRGZ9NrpT\nyBp3NBAEzwKmu+cxnNeBvIdxEOarinnCJwsGOisj8AZUnp46B4I7Xju3QCVSStarcNM0zU0l\nqFZ4dxYOZVxXeoDbBCv8iohT3TkQ7DCQNQPPFyLdn62ZIwI3v5aRexAKMMMa7TKiBGNN1fMg\nSEVFiEDhQMgSpCNWLigOQSymYiHqnJaDEJ+sWHRrLTUnYx4Vq2+ry5QsYiE9k1DwgkkclxGC\nj9WBGxHK1Vnd2RedigHXth+RWQw3ghxtuBIdpajtiX7MxXhjbTN32SeSWg54CdGSoCECW2TK\nmYh/hKg9u+xTcgkS8tNAdiO58xA8vYrwe1FW0twSxvbsTk0599XBfxEEUWpd9inaUrypVvno\nGGjdDAoC+VGvW3b6ON9eiBxCM10zGZ5k41vKT/BAlFD449p7RrUAwONPLTnUFdvdPlehXIBo\nV/ACQhiwlkc7YWlhPJEiMBNA9I4raalaVZKHlAWfqyJ6fhTYGRNCMIIlctKETYfAlGfb/RDK\n4vElNT8vbQuf5ejjHqzGTc1675zVFzfOAWzgVqO4VaRC+VTKK6hI4Jry84EwZ90zZ2PUQJzL\nOe6cfQHMTJih7p7h2aXMCxBmVl9ElOj7iSGzKAKn1062CBa5FRmJvqnB+CAIF1wXEdZwBQmo\nfszKmy1If30IPCkQ730tFT8UpP7azwA6MwnQ8y3ERsoOe2ouuAB4juCp6+uAI7MXyetEEOeS\nDs0hiiWUrblM5IwHUB+AQOR15qhUC9RoHItelvY0RRDkKPiL5jA4JOfBgZQEiHEUXPiUbVl4\nbodkth4rPRaB0vmbKz3Kao+8T2pAWbce8SW0NpGM/5qMGZa5nR/dWPpxftZc6QItVkCV+ThX\nBxwX/Nwv2Fff0sUACYjXRSgGOT9zPnmvS4oI4uAXqkOQfHW0Mc2ay0NAnNKyNu1QmU+oDeRI\nWplP1Bvu4nHbrgIvSy7hQdCBMlZOT6j8OJNAGS0fMvxOyBeWTFp7bxnqM1DJp+NqEBDvILGy\n41wjdsa0JGVhx7mLCFDoLm59/yr9cce3Voggd34LU3lBuOheJjDURdP5xr+qPJ0LHxmLK73Y\nMT8Az7442iaCNImCqJ68FheCgGhOvhqqNQ6JJxJ2AGUJAWDtarB8Y5B48xrVGueqC7cFahLo\nTNR0SLYiuAV7KwoYoVjjnKPEE0LaShMrCGXKcKt84yKS6VT5xgEZhVTxBj57NYY1znkA7YnY\nbCWtgjQt1LdSaEuL1yCYzAHac2aIX2nPJLJVslFa5RrzMtr4Wnl+pgrvQDzNoy4DpgEb6Ja1\njogEQHOuDYswzqOpK/1/V2GANBlOF2GUOlPUUYSBcyHiVaXCrLk4d7A+G5ZbYSwQ+6qswsBA\nTyo4FpMQ/hoveRqSfv90hctKfdbpLLg4QsDwVRJMwCAOOrGYApag3i70aSynOOdmFtcygYkH\n8YIbxRNnoitIPLPisZziInKmj+spSpm5mHc9RWG8KE9GFcJUvJOcO4WBnoxy/KwS6SSzKgLk\nyXvcOiWIIxusbzj3uNDhuYxg/YC8p8j6BvzQ29MIQgPHbSv3k3TBYoZNAgG5hBDyLEhB9lPf\nNJkgjiSovqEgPLP8mQqK+bwqhWhrlVaQMD7yS9CAu6djyWKGTeKAMosZzm+4I7cdWcyA235H\n7mgCQYvvlo4c4iLQ4rtlCIVlDptEwhIsc5gYqDoqB4Tzlrv4/CBYYfjbAmfAnR5auGABD6Rw\nKQsCSwbxoGYFRsN3MrAHcFSN2r39GdZ0h7WKYJ2HuDOLNViYEB+7WbCDVQjzY+cG7bm1XFTs\n4qkKAC7Q5n7IZYJ9A0wPYxngFq5tIwAABcPkRNEDONe8tsUsVMVwfaz1/BhVMaylgFGwhiE+\n1nSaIQCkFs6Dz6q0ycWI1WUC5VpZNkZwFjgriwoAEG9BgWv4M+KK8O16HoClCPxRnTUYBFnF\nTgzB+ROd7jDAXLTuZ4igUsG/X74uJW+vdJeCRRMbtdIKZQAgQjmXzR8A9GdmvkvcXQFGLuOW\nCZz5ObyVDYCgxXTtMz7Dj5kuIohb66PZ5POZnPl8tkdyOwV+Nu8Zh0s3ZmYGAECmZ7GrDICy\n4alMUQG4Aoiq9AR4yPN2hgQARHZsJ9AHfPPzJ4SuSr1MEHMYy1tUABgOAbjqz7AZCOP516qK\nA8HAthLgASBiaKVTDcfoLnkBQMo1oqBFF6J9cIRlrbi6ACbSWy8Xg4LM8e4GsE2jeWUHAK/z\nyaINlmGsD8a5i8Eu56Yhpj5akiMKQ2myAJvDwfeqxQDijuy82Q3w4BHxb+0ygXRjJ6EPA4h3\n314FAxxtx25IngQB8r5d4hUoiUCqkfdmTI6892WNYYnE+crUyg2fMf8wYLIMYFvgIk5GUEAQ\n28Pe08zvYMsMOU+0qgC4D304IRYAc0UfjjhFKRJ/JCJJIUqhZ9WzYBcA4t9z5geA/PcMLQEg\nCNdDQScRKMCTzgOA29szpz+w3XEMPTN3EmAuRw6OZIZ7JEfOkHMzwgQbkNhnHdOg46NWd/x4\nJuJenSoSLuBQts1lAhVBZEjWrkiasOXLvWcAbDPBJ751HY1BvZ7prgANhRS9OLsCBEE67m3n\n5/ON9JnxGZH0fivTIor8w35nrlfQd/jwtjw+B+NtPXM9ARBn6bdWdOESDsSqdhgEorWxlfES\nrteInZa+qDIKyTaWhmAoL7brmgCgLUpO0HWpJjt2muWiqHVsG8PCRBB+lg0u8hmZ8tANmOER\nGV8AwSUxNycHPdeMz77lqriIlQbFBRcAXYbMBRcgpRpAYWI5JxAAagAw/RkKFCtt0AHzxoUt\nR6xJVv8eIBT+zXdgF78ZU6mMOullAhP/zXXB9faV+zPu9jc/djLwEcsVBABnPuF12b1wVUi8\n7ImKQmKn6UcBSPNd3waYrAG04A/Wfww9uWmAtMdQXrFA18PPIZCsg1QsGeCyFOcOZQeDbO4f\nQeYst3rsT44XAAIXTxoYwXlA/c6U7nABx5M7BoA5qGfqMwD1KxOmw0uBfqf7wHoOGIvcoQDB\nXhV0Mr8CDSsu6Q+sL47rilQTGZN6S8OKulCYsHFLTh5Va17kkMjismhE1kMPhjUjtEnasQOY\n2OntmTwRLCKhldJKheA4sT03baLSjvTmwEe4gASmsfrK5BX0LO4hOItAmFOJGWtFNg3ucwSi\noDDJOehE4BvhhJJfgYzCrmsOQk5Bw2fPjFilnWmju58KPiNAxC43mikPweII84tsJZd6mpIk\n3CgaaZySFO0DwH3oI51cAOQkcKYz4CyWa7RwBcmTfRyV+SycPCW6ByC0wgQKSf8hkHKnQghg\njPU8St1LpDjcOQYuEekLO89yZOuCH+DZk8UkG6DlEVuOgox21fwHZ0M+HMC5t4NCd5k0+SMq\nUgOAoIxnhmIRyciE0ahKoRvNzXoAqJXPPk5Ulec/ZZMA0Eruk3cDKCEcNP/YzolmsB7uMsEj\nG90pUMGik4Db53mwMnEQu7vyiqrSm7Fz6ytDfQncsZVuAApMjiEd6xEHlZeM5VyHcHUJdkR9\n11VcIqf2MoEWwu/dOQiUDptmsmwsLQn4zp6DkM0j71pxLQA66zPreaLSsQJQNAXgTJ8zd7aD\nhSYdbr89JaQRHes7++ORVQX9Z3f4J1hlMrGWqM93oGMzM01CRSZzpKlnQcn5nXO6yhsEWjaX\nl+oAjZ/tfDLcs7DqseeEtKijY4sVbZcJlAyLqRwCLhT2L7o/w+vGzoRcrYrICtZvSroIlqNg\nYfkSss2I1gpv3wWrUQLANtnVKKs/T0GOBsLpKkQN16IgNr7yPLDfCGDbFm6q1Mokrmg3hWk9\nigwA+ccqeOUhWAAxGNsNYCnQY0fPEllqg6tzPSeWrhzAxfFlAncXiQ0yhqxT2YgC5Hm1U7Wz\n3wrBUardrf2qWrk+9sioAPLljlQj6lT9eTI4oW3SaHI6FXjBZ+WLlvvZpAwWrGzGQKxlCC0i\nVHzX17cGoz2RsxLIWiRIILqEGOfCYjamD8K8hACQ8iDDNS2IEpU7CeJu99Ka7CKBVhRmEPkQ\naAWCWPUZhjsW5U63AWQrGqYo7hUsaVE0zM49S1oQDavOsgPhLkapLg4J5EyeyQuBNoVVLqLN\n8NwtdWEdjAJ2CiQ0lR0gxJdPThFcEMkysjMRii4jHT/XxSBSqC26cFkMiGW1qaVCeRLbQoUy\nCkIqqS+aYuKl7LSjTR0vEPGUMW5qfIaoqOYV1tIocMpEdyFuj9TmtHwSBPhrVkyAIKiNEsOS\ngEHt2jM1OVheo1CvigtBGMSGiZSHii1hBLFhV3zfh2LYVYvnS4jjbO9wh0twGLLWeh21NIOx\ncIvKoNYhXL5z2IX6LoTUPbUjBZex8WwRFG0qXN2GEz1AGHdmTnE1QXS0cAu8JUHYtu2c8Fk6\nM7iVEM8xjEZHcf8fEOy+XkAjCXaKoqYzAoC9P+xa+FcsaS2qG/yZm45IY1fOfLDcpnPLxEsg\nlNsgNo20fuujKslKPBNsU7prCaYr5UDcApLbg/0NKDVcdj/x7R2d6p0lEAhw6VlkTQIDWnp7\n9GorXN2zwQUIo9U9O6GBUK17z0BpY+HfxV2raDk21bpP51QEym94fldmEOBUO0MjrMc5PxPx\nFiWLBLd7gltmdoPw1BCuHrlzQLIItEcMAHenDGUrX0IMPI+eyzHW6JzTI5rFnxUKLmPjT/v+\nINS+4YXQRURdm3d6R6rRwZ6ibT1rdBb3HdUWDoSKNbN9wwWEEAGQI6moGoBmzZHzDEQNGoD8\nkJkEPwhbrJq/LiBuBjkfQgctbN6WjNtRQLHBq7RQkc2N42elocqeS3u+PRH3grC93hJgK2h1\n21Ps4GEnaGXyMwgVEtO4AxxQhspda8e/WfuDy8kaxnB1BoiNLmt/8KueDLJgnQe3yFU+QACl\n2CXdH+gZ7Ol2AjsAt4t2c3yORSfYIcSOlZ+fCn2QGeBYeqgLGFMMJMyhTncgnqPCzR3rnaXP\nQNCvelc3mCNh0kN1mmJQ7RdzJ+wXBne5LyLlexAx6WE9j3TQWWWmhmwhCnw6QYqPmnBetTxh\nYpcFgeyV30J6R3nWtaF0ReSNVAvLoO4gtyR3AWCHkB1Wsl4JZDBn5s7QYEymsGLxuf2ZGaxV\nWxtUncn4GhJt7J2hXgeZcTWcYxau1wHJp6zMcObwrK2BVMJTWdeeBzVez7B/hQIeHDEzKsL6\nnUVi7xqBGeyY1HY7GycYyypMYHKQPFQCg0woNSACYaJpm5+Rh3CmQIrVfS+jrZwrlS1E15Ym\nkrc8I3blqSLBy+4tYoSTaWHqaxQIYCD15jwRtYEAYeJn7IyMs16mkzgU6h185LLVPXMgWHDm\nwGkGRHUMssJeAVNME5C5npVbIEziVKXEZcSUzSwRJmHKZl9pAVAFiWyl/izsGO/YTApsSm8j\nQspmf+bkru0/pBJ6sYPoLyQIdn4mYJJi5bO4hGCykaTo5Ytrapjs2A0gK0MdaQUQ961crTIL\nN7pa7yDV0qdSqhWyMfOHKliClE1PMiiGRZI0t3aWyUJfOCDre1cLMeSQqqw4XC4DYnepKyWK\nyal1mTBKgJTW5vN3pUAjNU8+J4hybNsDKJmYHKRNrJYZ6NP6RJO7VvVM6JWV6Eqgq0o8M1FS\ndEu3HdP6ma0uJiHL9XIBDUg+Z9bG1NdkwNqYpeaysj8o7mZO9FTlF9B0UvR0c4pwLgyztOX+\no5J8qCfsDANlQCs8eAkxGdSJOCJMeH4t9lh609XidOUxTBhl/lbkQEyC3s09QKKzgy1T3eMB\neKo75/NQvEkZ82p+GKrYYcNOhS6x4wOF2j1jICjhQVL0fvbPXNTDhH1P8F3JU2xR6Ru9pD1O\nPxBhwvNeuYfmaBYLEShmV3R3wkWNg5V+o0vJnQVE0d0HF63slgEbO9xZa0bCeqbXGjPLc+4s\nbIssz7kj7zxrcTbrO+zcsNAm1KfOgQVFz1BK4vt8AJubMJ3BgJ0e7p0GGhE2VDhh1puKKg93\nyy0l52E6j5vEm/I0M1PlOHkI9Iudv4LF15E1PM5mjadeJ0vxwp1aWAikdbx7wmRXKtx2bFIu\nlxQpQM3am63CJDmaw81yEcwfORDbOpTcwD4DuTUuguR6yMOtceudntTQJhqI8nciq3Fqxr85\nENv31JL6zfIblZd5G0rNv9h8R7LBepylzjqRhDFWds3xDQn1Lqnh9k8gbAZRn+DQUG/c2p8f\nH+5dglCkxcWdcOuwFXf/SPaFkaHPih0H50TYhacu7dIJqcbQfhxLbwaJN2mwHoCEc62qK3Zr\n2qbYO1R5KIoH5C2m4cazjLhUE/ZsaNnYiQRlfU1J+h4Iuxes1NRs5FIbtvqYOdBWUagKWWKo\nYJXVpc1APXZaTxd/qEMHe12sh6BYyl0rRNihoY2MSAz2gr1YEvsMzWaw7XVX3Qu2PYkTw71g\n20xjM1T/frFsd5ng+bUnHjLcCrY9+SkuvWGBsHwiF9qwytiLBxfasBRZpp+FNt0FzHl2Kk/c\nmbkwtOfGbg6O6g0uOlkt7d++1R4inp2koTAimzf4orcaqYQsySXEvilR3Lgkhhu9Rs15b7jR\na9THIG5X3FcmeXgg1dzXdPtZj6NadW+hDbeAjeb+8zHdAjbkQ19GrBZvubGCRJGmCvvX19h+\nOrLXJsh0OT+u2gOxb0RkAhzJVF8AB62nknLUnoCg8GIBmK1zEbEfNRYtsuMgqCwMFwACwPrG\nyLXTdA/YeMK/cLTY2WFm8tVUPwt2Cxg5DLsMYWUhdWJljZo9+NSVqqvODpKgqdmf7R98e6oa\nVHBrNgn1O3YaZCbsQFlipw8/69OyQk0jQNyywrrCkh2NY59sqjsc6/edmgO01IzDOz/T7WX7\nnTfMzWX7nfsYcDuLmmogGHMJsdy/PwKsaiDV3o8cSI1bsiI4lL2kaviyciDqvDuRgIR7uVSn\nLIKolUt1JBeEjUjUHZ6SIG+I1fA+mTrIsOGKzO3UPMAidrnRk/2+WbJOp+gK1wexZN3y0tWV\nJfvIAFDDelZNgKgHS7bRDyR2IdeczXeUs6AKscZty27AWvo+ckGMmh62U8leQeEqn4s9hixm\n42lDlKIw3XJl5o4okt6X+iI5kDxV0HyxedJOxB64fT4qpy5ObMLk3+EeuP1J5ZsqSrpYOO7n\n4wa3cEOtqstdhVYuBlT4wypxmVJU+fhL24v/uS1nuTZilY96aXkROpWVyJrwncewpwRczmHr\n4pazfec9W7cKwhVcWsqxYrG3PA5k92lUO9CqArpc7N2N1Cw5C3BB1D38zow7L3VAbG24GxYa\nyKZkFTcGvzMNzwmJ7N6m24VCSbdvc+xvsckdS80dB1nKEGDjOknmKu7j/SQN4d0jtX3T7i5c\nKXSxkFwWaFW37c72e7Gq+3aX9DlAthv79SRDV1TSTVtVEj0eb4+1QboiLctWc+NulwPEyr61\nNZvUx8rGtTX3ThbbTahX4jZQ2+cnmORSIZIlI6VaIZao1zz5dHtJL2RdPcSqdZl1lKZkw8nB\n3HWgmg0nfdHhHs81Y0crG8pm185Y2VBWua8c6NVR1vtGq7vvckvz58KgZhdcZLozbVXD7WAp\nkHqN2tHH7mZ1e1QZJVb+uKmqRWi4Q7KjYhdRK+7Xaqka7hLe0mqzGMjNYbUGRDHQrat2iQ7R\nKq633z5ou1tt3qNpwW85ZyFSe7vHLUvbSPQly8tUW4nR0nVjfZDb8t45LrtKZKPeYHnQzoa/\nuppX89pU8WwxGxm4c6kPidwblProfT+RAXZX/6jzcn6N3YFG1obGUjfA9kSGWPvjXvDyG5Fv\n7fbROw/JLsuWxH0/XZa9iMd7YdxlOZMgWcalk9uLBnHfZfufLiF6NdwORy7cpjsHchvqnoc8\n7cd7Dvy0H9e45WkRrlX0ReQm4c7n2tk8NjIys7N57JMDtLNXbOaOcKDtTv2ONu7qDsnZ3ilc\nWvRqVh+uLSLpWsqilKi4Ff30MdkG3yssvlFmPg0dRCTyT34+su6He4nbcdvZHDbb2YFkw29b\nbZcS6WUCBtgK8usFdEw8/fSdWbQlbdk8QqT6JQBO99zx6W7nW42WQdRF2S9bAMjW4s582JGt\nxe0kuQChWXTi9Z4sx5d3d7vkfCUFSL7Jy0Zsd7dLzt5/wdIivxPH+ZiqLXq9wyPYBCLfF5KH\n+BUwdoS3wtF+J81lpO7I+Zqa2Ezher0AJ1zK83prTrhwJ9/Hcxnl625ajjPy3TYW3WlB/Zbk\nq84gZx4oX3XmHdSd7VrzXUggZXwB+ZIwOoyX0O4vRNL9Nja52Czd2d+DfAOWrdN2kuj3aD2v\njsqv7efVUSLb3ZHzPVUk+SqbFyp+lY0dP7yvpH8B+Xo2G0dW/Ay/leqF/F4Yb0yh5mfUL2T4\n7TZe6bLmZ2ugb9DOd4vlJT7vTCPpqPDxy2zU2gCk3DnQC/mVSbLgIPkqrxeJ6muk2He+y8Rv\n6Xmh0fOgafK8jO5F/F4af6k8KqZMKpDnZXm0EZ31Q/MLeb3nbZnkW9zUnaqzYijfPFd9zM5X\npL3Ity+n664QesgltPJleT5m5DAPyLfghYF6lr8IEn2+fQcfiLoqf0vUVTmLv7pLdr4nxQKl\npLDOAhzfnBdZvjm66f0lYQkeAZu+6+P1Mr3ZTFbJb+lr6/UWvBdJaVZ6dkdLhhybM8xfX9hJ\nXnEEEa3sJuwdwssFtpxvcix4kyMXLFgfF2XZ/fbd1+rd1+rd1+rd1+rd1+rd1+rd1+rd1+rd\n1+rd1+rd1+rd1+rd1+rd1+rd1+rd1+rd1+rd1+rd1+rd1+rd1+rd1+rd1+rd1+rd1+rd1+rd\n1+rd1+rd1+rd1+rd1+rd1+qXd1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ArNF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","text/html":"<img src=\"https://cdn.kesci.com/upload/rt/6A9F322BD84142EA8E1B9CA6B598017D/t4krdza83q.svg\">"},"metadata":{"image/png":{"width":480,"height":360},"image/jpeg":{"width":480,"height":360},"image/svg+xml":{"width":480,"height":360,"isolated":true},"application/pdf":{"width":480,"height":360}}}],"execution_count":14},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"F684ACBD4244485CA6E758A65C6C8B67","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"### 2. 通过 HDI + ROPE 进行检验  \n\n**ROPE**（Region of Practical Equivalence，实际等效区间）是一种贝叶斯统计中的概念，用来定义在实践中认为与虚无假设无显著差异的区域。通过结合 HDI 和 ROPE，我们可以进一步检验成功率 $\\pi$ 是否在实践中显著不同于 0.5。  \n\n\n在这个例子中，我们假设 ROPE 的范围为 $[0.4, 0.6]$。这意味着如果成功率 $\\pi$ 落入这个区间，我们认为 $\\pi$ 在实践中与随机判断水平 0.5 没有显著差异。因此，我们的目标是判断 95% HDI 区间是否完全落在 ROPE 内，完全在 ROPE 外，或者部分重合。  \n\n> 注意：ROPE需要我们根据具体情况进行调整。 一般情况下需要参考经验值或文献中的建议。"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"549433CC837645C9A5ED1C6BD794836B","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"**检验步骤**  \n\n\n我们已经知道 95% HDI 区间为 $[0.76, 0.89]$。接下来，我们需要将这个区间与 ROPE 区间 $[0.4, 0.6]$ 进行比较：  \n\n- HDI 区间为 $[0.76, 0.89]$。  \n- ROPE 区间为 $[0.4, 0.6]$。  \n\n接下来，我们考虑三种可能的情况：  \n\n- **HDI 完全落在 ROPE 之内**：如果 95% HDI 完全在 $[0.4, 0.6]$ 之内，那么可以认为 $\\pi$ 在实践中与 0.5 无差异。  \n- **HDI 完全落在 ROPE 之外**：如果 95% HDI 完全在 $[0.4, 0.6]$ 之外，那么可以认为 $\\pi$ 在实践中不同于 0.5。  \n- **HDI 与 ROPE 部分重合**：如果 95% HDI 区间与 ROPE 有部分重合，那么可以认为不确定 $\\pi$ 是否不同于 0.5。"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"C8C0DD01073643FA97C73BE28F7E54D0","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"**结果分析**  \n\n在我们的例子中，HDI 区间 $[0.76, 0.89]$ 完全位于 ROPE 区间 $[0.4, 0.6]$ 之外。这意味着我们可以认为 $\\pi$ 在实践中与 0.5 存在差异，且成功率 $\\pi$ 高于 50%。  \n\nHDI 区间 $[0.76, 0.89]$ 完全落在 ROPE 之外，这表明成功率 $\\pi$ 在实践中不同于随机判断水平 0.5。结合 HDI 和 ROPE 的结果，我们可以进一步确认成功率 $\\pi$ 高于 0.5。  \n\n因此，基于 HDI + ROPE 的检验结果，我们可以认为成功率 $\\pi$ 不仅在统计学上（通过 HDI 检验），而且在实践中（通过 ROPE 检验）都不同于 50%。"},{"cell_type":"code","metadata":{"jupyter":{},"collapsed":false,"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"24184F5901854CE19A6C751413626313","notebookId":"68f9c684c011c75553f024e8","trusted":true},"source":"pp <- p + \n  geom_segment(aes(x = 0.4, xend = 0.6, y = 0, yend = 0), color = \"green\", size = 2)\n\npp","outputs":[{"output_type":"display_data","data":{"text/plain":"plot without 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kvSwlhRGf2neO1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\ngqwhEN7RjYoCTd3wPSw0X0+P7G5Mr67Pv0h3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LjE1MjM0NCA1NTYuMTUyMzQ0IDU1Ni4xNTIzNDQgNTU2LjE1MjM0NCA1NTYuMTUyMzQ0\nIDU1Ni4xNTIzNDQgMCA1NTYuMTUyMzQ0IDU1Ni4xNTIzNDQgMCAwIDAgMCAwIDAgMCAwIDY2\nNi45OTIxODggMCA3MjIuMTY3OTY5IDAgNjEwLjgzOTg0NCAwIDcyMi4xNjc5NjkgMjc3Ljgz\nMjAzMSAwIDAgMCAwIDAgMCA2NjYuOTkyMTg4IDAgMCAwIDAgMCAwIDAgMCAwIDAgMCAwIDAg\nMCAwIDAgNTU2LjE1MjM0NCA1NTYuMTUyMzQ0IDAgNTU2LjE1MjM0NCA1NTYuMTUyMzQ0IDAg\nMCAwIDIyMi4xNjc5NjkgMCAwIDAgMCA1NTYuMTUyMzQ0IDU1Ni4xNTIzNDQgMCAwIDMzMy4w\nMDc4MTIgNTAwIDI3Ny44MzIwMzEgNTU2LjE1MjM0NCAwIDAgNTAwIDUwMCBdCiAgICAvVG9V\nbmljb2RlIDEzIDAgUgo+PgplbmRvYmoKMTAgMCBvYmoKPDwgL1R5cGUgL09ialN0bQogICAv\nTGVuZ3RoIDE4IDAgUgogICAvTiA0CiAgIC9GaXJzdCAyMwogICAvRmlsdGVyIC9GbGF0ZURl\nY29kZQo+PgpzdHJlYW0KeJxVkc1qwzAQhO9+irkUkkssyX9tMDkkgVBKISQ9tfQglMURFMtI\ncmnevpIduxSd9mN2Z3bFwRJeomCJAM/LhFfIyqekrpG+3TpCepQNuQRA+qIvDh8QYDjhc0A7\n07cePNlsho6jNZdekcVCSW0N+Io/rnIsrt53bp2mA22s7K5auZWxzXI5jrEkvTbtXnrCYr8W\nTBSciYxVRSnE+3Ka/5cID8E1th6lpRghhhrAK1203JqfkJSFV1Ql8kzMeVsf5A75rD9Y03eo\n61jEevQY6ITOgVrZui56qduEn+FtT1O1C6o9fWtFp8M2wpA58hM501tFDtnseQ6Nyo/RXfiA\nf+vtpJdfprlvF45/Xy6IfgGfEG4oCmVuZHN0cmVhbQplbmRvYmoKMTggMCBvYmoKICAgMjc2\nCmVuZG9iagoxOSAwIG9iago8PCAvVHlwZSAvWFJlZgogICAvTGVuZ3RoIDc4CiAgIC9GaWx0\nZXIgL0ZsYXRlRGVjb2RlCiAgIC9TaXplIDIwCiAgIC9XIFsxIDIgMl0KICAgL1Jvb3QgMTcg\nMCBSCiAgIC9JbmZvIDE2IDAgUgo+PgpzdHJlYW0KeJxjYGD4/5+JgYuBAUQwMZoqMzAwMvAD\nCVNukBgHkBVSAOLuAhJmNkAinAHECgQSgYUgohNIBLuAiGiIKYwggpkxAiQRsYCBAQAP2wqx\nCmVuZHN0cmVhbQplbmRvYmoKc3RhcnR4cmVmCjIyNjg4CiUlRU9GCg==","text/html":"<img src=\"https://cdn.kesci.com/upload/rt/24184F5901854CE19A6C751413626313/t4kre16nx7.svg\">"},"metadata":{"image/png":{"width":480,"height":360},"image/jpeg":{"width":480,"height":360},"image/svg+xml":{"width":480,"height":360,"isolated":true},"application/pdf":{"width":480,"height":360}}}],"execution_count":15},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"E782075C76984B54AE14086B12FB7D19","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"### 贝叶斯因子(Bayes Factor, BF)  \n\n贝叶斯因子是用于比较两种假设（通常是虚无假设和备择假设）相对支持程度的一个量化工具。它通过计算在两种假设下观察到的数据的概率，来评估这些假设的相对可信度。  \n\n从先验到后验，**数据**让备择假设的相对可能性发生了改变。  \n\n我们可以使用**贝叶斯因子**来量化**数据**对我们信念更新的程度：  \n\n$$  \n\\begin{align*}  \n\\text{Bayes Factor} &= \\frac{\\text{posterior odds }}{\\text{prior odds }} \\\\  \n&= \\frac{\\text{后验概率比}}{\\text{先验概率比}}  \\\\  \n&= \\frac{P(H_a | Y) / P(H_0 | Y)}{P(H_a) / P(H_0)}  \n\\end{align*}  \n$$  \n\n\n* $BF_{a0}$ = 1，收集到的数据并没有改变备择假设$H_a$的相对可能性  \n* $BF_{a0}$ > 1，收集到的数据增加了备择假设$H_a$的相对可能性。BF越大，表明支持备择假设$H_a$的证据越强  \n* $BF_{a0}$ < 1，收集到的数据削弱了备择假设$H_a$的相对可能性。BF越小，表明支持备择假设$H_a$的证据越弱  "},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"65E3E7BDC9AD4FDAA0017E1DE173CCA8","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"|**贝叶斯因子** |**解读**|  \n|-|-|  \n|> 100|极强的证据支持$H_a$|  \n|30 ~ 100|非常强的证据支持$H_a$|  \n|10 ~ 30|较强的证据支持$H_a$|  \n|3 ~ 10|中等程度的证据支持$H_a$|  \n|1 ~ 3|较弱的证据支持$H_a$|  \n|1|没有证据|  \n|1/3 ~ 1|较弱的证据支持$H_0$|  \n|1/10 ~ 1/3|中等程度的证据支持$H_0$|  \n|1/30 ~ 1/10|较强的证据支持$H_0$|  \n|1/100 ~ 1/30|非常强的证据支持$H_0$|  \n|< 1/100|极强的证据支持$H_0$|  \n\n> Source: 该表改编自胡传鹏等(2018)，源引用于Lee & Wagenmakers (2014)。"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"C204283237AE4E1282F8E28DE70EB2CC","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"### 3. 通过贝叶斯因子 (Bayes Factor, BF) 进行检验  \n\n回到之前的例子，我们已知$\\pi = 0.8$是成功事件发生的概率，先验分布为 $Beta(2, 2)$； 后验分布为 $Beta(101, 21)$。  \n\n\n* 但此时我们会遇到一个问题，因为$\\pi$的后验分布是连续的，对于连续型变量，单个值发生的概率为0(一条线)：  \n$$  \nP(\\pi = 0.8 | Y) = \\int_{0.8}^{0.8} f(\\pi | y)d\\pi = 0  \n$$  \n\n* 那么备择假设发生的概率为：  \n$$  \nP(\\pi \\ne 0.8 | Y) = 1 - P(\\pi = 0.8 | Y) = 1  \n$$  \n\n* 此时如果我们仍然沿用刚刚的公式计算后验概率比：  \n$$  \n\\text{Posterior odds } = \\frac{P(H_a \\; | \\; Y)}{P(H_0 \\; | \\; Y)} = \\frac{1}{0} = \\text{ nooooo!}  \n$$  \n"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"9A3B67458B4341B5B2E405A03B3D1131","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"既然在连续型变量中，考虑单个值的概率会遇到计算上的问题，我们需要引入**区间**来计算贝叶斯因子，比如使用 ROPE 的范围，检验成功次数比例$\\pi$是否在(0.4, 0.6)这个区间内：  \n\n$$  \n\\begin{split}  \nH_0: & \\; \\; \\pi \\in (0.4, 0.6) \\\\  \nH_a: & \\; \\; \\pi \\notin (0.4, 0.6) \\\\  \n\\end{split}  \n$$  \n\n**贝叶斯因子假设检验公式**  \n\n$$  \nBF = \\frac{P(H0|Y)/P(H0)}{P(H1|Y)/P(H0)}  \n$$  \n\n* 注意：我们需要为 H0 和 H1 设置先验概率，这里我们假设两者出现的概率为 0.5, $P(H0) = 0.5$, $P(H1) = 0.5$。  \n"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"2FC79E8969FF471E9F884CDEF0AED77E","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"\n**贝叶斯因子的计算**  \n\n\n首先，我们需要确定 $\\pi$ 在 ROPE 范围 $(0.4, 0.6)$ 内的概率，即：  \n\n$$  \n\\begin{align*}  \nP(H0|Y) &= P(0.4 \\leq \\pi \\leq 0.6 | Y) = \\int_{0.4}^{0.6} f(\\pi | Y) d\\pi \\\\  \n\\end{align*}  \n$$  \n\n- 其中 $f(\\pi | Y)$ 是 $\\pi$ 的后验密度函数。根据我们的后验分布 $Beta(101, 21)$，我们可以使用积分或数值方法来求解具体数值。  \n\n与此同时，我们考虑 $(0.4, 0.6)$ 之外的概率，即 $P(\\pi \\notin (0.4, 0.6) | Y)$，通过：  \n\n$$  \nP(H1|Y) = P(\\pi \\notin (0.4, 0.6) | Y) = 1 - P(H0|Y)  \n$$  \n\n现在我们可以根据下式计算贝叶斯因子（BF）：  \n\n$$  \n\\text{Bayes Factor } = \\frac{P(0.4 \\leq \\pi \\leq 0.6 | Y)/P(H0)}{P(\\pi \\notin (0.4, 0.6) | Y)/P(H1)}  \n$$  \n\n**由于两种假设下先验分布相同，$P(H0) = P(H1) = 0.5$，因此可以简化计算为：**  \n\n$$  \n\\text{Bayes Factor } = \\frac{P(0.4 \\leq \\pi \\leq 0.6 | Y)}{P(\\pi \\notin (0.4, 0.6) | Y)}  \n$$  \n"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"700E0997B6264ECBB2A414832C55E167","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"**结果分析**  \n\n\n计算 $P(0.4 \\leq \\pi \\leq 0.6 | Y)$ 可以用贝塔分布的累积分布函数（CDF）进行:  \n\n$$  \nP(0.4 \\leq \\pi \\leq 0.6 | Y) = F(0.6) - F(0.4)  \n$$  \n\n- 其中 $F$ 是以 $Beta$ 分布(101,21)累积分布函数。则，  \n\n$$  \nP(0.4 \\leq \\pi \\leq 0.6 | Y) \\approx 0.022 - 0.0003 = 0.0217  \n$$  \n\n因此，$P(\\pi \\notin (0.4, 0.6) | Y) = 1 - 0.0217 = 0.9783$。  \n\n计算贝叶斯因子（BF）:  \n$$  \n\\text{Bayes Factor} = \\frac{P(0.4 \\leq \\pi \\leq 0.6 | Y)}{P(\\pi \\notin (0.4, 0.6) | Y)} \\approx \\frac{0.0217}{0.9783} \\approx 0.0222  \n$$  \n\n贝叶斯因子的值约为 $0.0222$，这意味着备择假设 $H_a$ ($\\pi$ 不在 $(0.4, 0.6)$ 区间内) 相对于原假设 $H_0$ ($\\pi$ 在 $(0.4, 0.6)$ 区间内) 获得了极大的支持。这个结果表明，给定数据和先验信息，$\\pi$ 值显著不在 $(0.4, 0.6)$ 区间内，大大倾向于 $\\pi$ 的真实值远离 $0.5$，这与 $\\pi = 0.8$ 的假设一致。  \n\n因此，我们可以结论认为，在给定的统计模型和数据条件下，成功率 $\\pi = 0.8$ 是显著高于一般随机判断水平（$0.5$），这进一步强化了 $\\pi$ 显著高于 $0.5$ 的结论。此结果对于决策支持和进一步的统计分析具有重要意义。"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"545C586427F74D859AE19A0FBB2B71CA","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"**额外补充：单侧检验**  \n\n贝叶斯后验模型可以直接进行单侧检验，例如要检验成功率小于80%的假设，我们只需要计算后验分布中$\\pi$小于80%的概率即可。  \n\n* 那么，我们可以考虑在后验分布$Beta(101，21)$中，$\\pi$小于0.8的概率  \n$$  \nP(\\pi < 0.8 \\; | \\; Y = 99) = \\int_{0}^{0.8}f(\\pi | y=99)d\\pi .  \n$$  \n\n(如图，可以理解为$\\pi$从0 ~ 0.8围成的曲线下面积)  \n\n![Image Name](https://cdn.kesci.com/upload/sm3ylkxvv1.png?imageView2/0/w/720/h/960)  \n\n经计算可知，在$Beta(101, 21)$中，$\\pi$小于0.8的概率是20.2%  \n\n\n<div style=\"padding-bottom: 30px;\"></div>  \n"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"33F1D3AB12AE4A309E90254F606F8AEE","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"**单侧检验--假设检验**  \n\n如果我们把这个问题放到假设检验的框架下，那么这个问题可以写成：  \n\n* 虚无假设 ($H_0$)：在随机点运动任务中，成功次数占比等于0.8（即80%）  \n* 备择假设 ($H_1$)：在随机点运动任务中，成功次数占比大于0.8（即80%）  \n\n$$  \n\\begin{split}  \nH_0: & \\; \\; \\pi \\ge 0.8 \\\\  \nH_a: & \\; \\; \\pi < 0.8 \\\\  \n\\end{split}  \n$$  \n\n我们可以计算出这两个假设对应的后验概率：  \n\n* 备择假设：$\\pi < 0.8$  \n$$  \nP(H_a \\; | \\; Y=99) = 0.202  \n$$  \n\n* 虚无假设：$\\pi \\ge 0.8$  \n$$  \nP(H_0 \\; | \\; Y=99) = 1-P(H_a \\; | \\; Y=99) = 0.798  \n$$  \n\n将这二者相除，我们可以知道备择假设发生的可能性大约是虚无假设的3倍，这被称为**后验概率比(posterior odds)**  \n\n$$  \n\\text{posterior odds } = \\frac{P(H_a \\; | \\; Y=99)}{P(H_0 \\; | \\; Y=99)} \\approx 0.253.  \n$$  \n"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"EE97C2E08D604B0D8F90A6B86BB81612","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"我们说$\\pi$的后验概率表现了对$\\pi$信念在数据与先验之间的平均更新，那么我们可以看一下在**先验分布** $Beta(2,2)$ 中，虚无假设和备择假设发生的可能性  \n\n* 备择假设：$\\pi < 0.8$  \n$$  \nP(H_a) = \\int_0^{0.8} f(\\pi) d\\pi \\approx 0.8  \n$$  \n\n* 虚无假设：$\\pi \\ge 0.8$  \n$$  \nP(H_0) = 1-P(H_a) \\approx 0.2  \n$$  \n\n* 先验概率比：  \n$$  \n\\text{Prior odds } = \\frac{P(H_a)}{P(H_0)} \\approx 4  \n$$"},{"cell_type":"markdown","metadata":{"jupyter":{},"scrolled":false,"tags":[],"slideshow":{"slide_type":"slide"},"id":"A57A24BBDDF0455F875680AF55C10B28","runtime":{"status":"default","execution_status":null,"is_visible":false},"notebookId":"68f9c684c011c75553f024e8"},"source":"\n> $$  \n> \\text{posterior odds } \\approx 0.253  \n> $$  \n> $$  \n> \\text{Prior odds } \\approx 4  \n> $$  \n\n从先验到后验，**数据**让备择假设的相对可能性发生了改变。  \n\n我们可以使用**贝叶斯因子**来量化**数据**对我们信念更新的程度：  \n\n$$  \n\\text{Bayes Factor} = \\frac{\\text{posterior odds }}{\\text{prior odds }} = \\frac{\\text{后验概率比}}{\\text{先验概率比}}  \n$$  \n\n代入我们的数据中  \n$$  \n\\text{Bayes Factor} = \\frac{0.253}{4} \\approx 0.063  \n$$  \n\n* 在这个例子中，贝叶斯因子大约是0.063，这表明，数据对虚无假设的支持度实际上是大于备择假设的，相比先验分布，数据使得备择假设相对虚无假设的支持度下降了。  \n\n\n综上，后验概率 $P(H_a \\; | \\; Y=99) = 0.202$  表示数据对备择假设$H_a$ 的支持度是20.2%，反映出数据并未提供有力证据来支持成功率低于 80% 的假设。这与频率主义统计方法的决策机制不同：  \n\n* 不同于我们曾经在频率主义框架下所做的那样：根据*p*值的大小拒绝或无法拒绝零假设。  \n\n* 在贝叶斯学派中，不提倡一刀切标准。数据可以告诉我们后验概率或贝叶斯因子，从而告诉我们证据有多**强**。它只是衡量不同的假设发生的相对可能性。  \n\n\n<div style=\"padding-bottom: 30px;\"></div>"},{"cell_type":"markdown","metadata":{"id":"C3ACC6AEE748437A80639554A8102AC1","jupyter":{},"notebookId":"68f9c684c011c75553f024e8","runtime":{"status":"default","execution_status":null,"is_visible":false},"scrolled":false,"slideshow":{"slide_type":"slide"},"tags":[]},"source":"## Summary and discussion  \n\n<p align=\"center\">  \n  <img src=\"https://cdn.kesci.com/upload/sm7veon5ei.png?imageView2/0/w/720/h/960\" alt=\"Image Name\">  \n</p>  \n\n本节课我们学习了如何利用精确或近似的后验模型，对未知参数进行统计分析，包括：  \n1. 后验预测(均值和可信区间)  \n2. 基于后验的假设检验  \n3. 后验预测  \n\n😎到这里你或许发现了贝叶斯的独特之处，相比于之前学习的基于频率学派(frequentists)的统计方法。  \n\n<div style=\"padding-bottom: 20px;\"></div>"},{"cell_type":"markdown","metadata":{"id":"6BFBC273F46C4031999295527A49EB78","jupyter":{},"notebookId":"68f9c684c011c75553f024e8","runtime":{"status":"default","execution_status":null,"is_visible":false},"scrolled":false,"slideshow":{"slide_type":"slide"},"tags":[]},"source":"贝叶斯分析结果的易解释性 -- 样本和总体的关系  \n\n- 贝叶斯分析根据观察到的数据 Y 来评估未知参数 $\\pi$ 的不确定性。  \n    \n    - 例如，在随机点运动任务中，有 118 次尝试，其中 99 次成功完成任务。  \n    - 根据这一观察结果，我们确定在随机点运动任务中成功完成的比例 π 低于 0.80 的后验概率为 20.2%  \n        \n- 在频数分析中，往往根据对于总体性质 $\\pi$ 的假定值来评估观测样本Y的不确定性。  \n    \n    - 频率学派假设只要无限(long-run)的进行抽样，样本分布的参数就会近似总体分布。  \n    - 然后，根据零假设显著性检验(Null Hypothesis Significance Testing)和 p 值来判断样本和总体的关系。  \n    - 例如，频率学派假设 $\\pi$ 为 0.80时，我们观察到的样本(118 次尝试)中，有 $Y = 99$ 次的成功次数，那么这一结果可能会被视为样本与总体的差异。  \n        \n\n假设检验的哲学基础是波普尔的科学可证伪性  \n  - 它所回答的问题对人脑来说却**不那么自然**：因为我们实际上观察到了数据，但却不知道π，所以要解释一个假设相反的计算结果，可能会让人费解。  \n  - 然而，在检验假设时，更自然的做法是问 \"我的假设有多大可能性？(贝叶斯概率的答案），  \n  - 而不是 \"如果我的假设不成立，我的数据的可能性有多大？(频繁概率的答案）。  \n  - 鉴于 $p$ 值经常被曲解，进而被误用，整个频繁主义和贝叶斯理论都越来越不强调 $p$ 值。"}],"metadata":{"kernelspec":{"language":"R","display_name":"R","name":"ir"},"language_info":{"codemirror_mode":"r","name":"R","mimetype":"text/x-r-source","file_extension":".r","version":"3.3.2","pygments_lexer":"r"}},"nbformat":4,"nbformat_minor":4}